THE ELLIPTIC INTEGRAL IN ELECTROMAGNETIC THEORY* · THE ELLIPTIC INTEGRAL IN ELECTROMAGNETIC...

THE ELLIPTIC INTEGRAL IN ELECTROMAGNETIC THEORY*

BY

ALFRED GEORGE GREENHILL

The analytical complexity in the reduction of the elliptic integral in electro-

magnetism, as well as in most dynamical problems, arises in consequence of the

appropriate integral of the third kind being of the circular form in Legendre's

terminology, and the elliptic parameter of Jacobi is then a fraction of the

imaginary period.

The expression of the integral by the theta function would imply then a

complex argument, and a table of the theta function would not be of complete

utility unless made a triple entry table.

When required in a problem of electromagnetism it is the complete third

elliptic integral which usually is sufficient for a solution, and this, as shown by

Legendre, can be expressed by elliptic integrals of the first and second kind,

complete and incomplete ; and for these the Table IX of Legendre provides

the material for a numerical evaluation.

As an important application we may cite the calculation of the mutual induc-

tion of two coaxial helices, employed in the ampère balance designed for weigh-

ing their electromagnetic attraction by the late Viriamu Jones and Professor

Atrton, and so arriving at an independent determination of the electrical units.

The present investigation was undertaken in the lifetime of Professor Viriamu

Jones, with the object of exhibiting the result of his complete third elliptic inte-

gral in its simplest form, suitable for immediate numerical computation, and

also to reconcile the conflicting notation of different writers on the subject by

adopting Maxwell's Electricity and Magnetism as the standard.

Incidentally the quadric transformation of Landen is required here so fre-

quently that a digression has been made on the theory, and an elucidation sub-

mitted of its essential geometrical interpretation.

The references in the course of the work will be chiefly to —

Maxwell, Electricity and Magnetism.

Webster, Electricity and Magnetism.

Gray, Absolute Measurements in Electricity and Magnetism.

CAYLEY,Proceedings of the London Mathematical Society, vol.6.

* Presented to the Society September 5, 1907. Received for publication January 29, 1907.

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448 A. G. greenhill: [October

Viriamu Jones, Philosophical Magazine (1889), Philosophical

Transactions (1891), Proceedings of the Royal Society (1897).

Minchin, Philosophical Magazine (1893), (1894).

Burnside, Messenger of Mathematics, vol. 20 (1891).

Coleridge Farr, Proceedings of the Royal Society (1898).

Nagaoka, Journal of the Tokyo College of Science, 16 (1903).

Coffin, Rosa, and Cohen, Electromagnetic Integrals, Bulletin of the

Bureau of Standards (1906).

Alexander Russell, Magnetic Field of the Helix, Philosophical Mag-

azine, April, 1907.

Notation and preliminary theory.

1. The elliptic integrals of the I, II, and III kind (abbreviated in the sequel

to I. E. I. ; II. E. I. ; III. E. I.) are composed of differential elements of the

formds ds 1 ds

(*) vs' {s-a)7s> s-=«7s>

(2) S = 4-s — sx-s — s2s — s3, sx>s2>s3;

but for analytical simplification it is convenient to normalise them to zero degree

by an appropriate factor, so that the elements may be written

i/(sx — s3)ds s — a ds \V^ ds

(3) ~VS ' V^-sftTS' s~^dVS'

(4) 2 = 4o- — sxcr — s2cr — s3;

and in the circular form of the III. E. I. the expression 2 is negative and the

normalising factor is changed to \V( — 2).

The same normalising is required with the elementary circular or hyperbolic

integral which arises when S is of the second or first degree, in consequence of

s , or s, and s2 being made infinite, so that we take

(5) S= 4-s — sf s — s3, or 4¡s — s3;

and the integral corresponding to the III. E. I. becomes

<6> J s-cr 1/(4.8-vs_S3)-tan Vv-«S«-«J'

m fv/(«3-^) ds_Is-B, js-j,CO J s-cr V(4-s-s3)-am Vs-t Vss-t'

with corresponding hyperbolic-logarithmic forms.

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1907] THE ELLIPTIC INTEGRAL IN ELECTROMAGNETIC THEORY 449

The standard III. E. I. adopted in the sequel is written

M-2)<«> /*£ vs>

in which the sign of 2 is to be changed in the infrequent case which arises when

the hyperbolic-logarithmic III. E. I. is to be employed ; and it is preferable to

keep to this form of the integral of an algebraic function, and to settle the

sequence of magnitude of

er, s, sx, s2, s3,

before proceeding to a reduction in the notation of Weierstrass, Jacobi, and

Legendre to their standard form, if requisite for comparison.

In an electromagnetic application it is the complete III. E. I. which gives

the solution, and then the limits of integration are oo, sx, s2, s3, — oo ; and now

the great theorem discovered by Legendre (Fonctions elliptiques, I, chap. 23)

enables us to express the result by incomplete integrals of the I and II kind,

the I. E. I. and II. E. I., on which a digression is made at the outset.

2. The jacobian quarter periods, K and K', are defined by the complete

integrals

m w p«V(sx-s3)ds p-vÇ».-O«*"".

and the incomplete integrals may be written

1-9A as- r'V(«i-«.)<fr fW, p-'V^-sJdc-(2) hK=L—78—•' fK=£„ •(-s) '

where h and/ are real proper fractions defined by the ratios

(3) h= + \ -p-~, /= I h- I .

In the jacobian notation

(4) P.S2 *3 7 '2 *' *•

and whenoo>s>Sj, or s2> s> s3,

(5) snU^=Si^8-3, or ~-3,

(6) cn2hK=---\ or

s„

*2-*3'


450 A. G. GREENHILL: [October

(7) dn2hK=S-^, or ^"—;s-s3 sx-s3

while withs1>o->s2, or s3>o-> —00,

(8) sn2/iT=^, or ÎL=A,

(9) cn2//t" = —-2, or ^^,V ' J sx-sf sx-cr'

(10) dn2 fK' = ^3, or *=£.sx— &3 6j — a

It is the advantage of the use of the fraction h or / of the period K or K

that the corresponding modulus k or A' is indicated thereby and need not be

written down.

In the notation given already in equations (2) the incomplete I. E. I. can

be written

and for the complementary modulus &',

3. Besides the first elliptic integral (I. E. I.), given already in § 2 (1), (2),

(11), (12), (13), (14), we shall require the second elliptic integral (II. E. I.)

complete and incomplete.

The jacobian zeta function which expresses the incomplete IL E. I. may be

efined by the standard integral

(1) -r?-. -4, = dn2 hK- Kdh = E am hK= hH + zn hK,

or

<*> £VW^) ~Tr£^fK'-K'df = Ean,fK'=fH'+,nfK'.,



where zn u in Glaisher's notation denotes the jacobian function Zu ; also

H and H' denote the complete IL E. I. to moduli k and k', defined by

(3) H= f dn2 hK- Kdh, H' = f dn2 fiK' ■ K'df,Jo Jo

corresponding to K and K', the complete I. E. I.

The letter H is used instead of the usual E, as E will be required for the

quadric transformation.

With Glaisher's notation

(4) zs u = zn u + -j- log sn u,

the incomplete II. E. I. in the regions

00 >s>sx> s2>s> s3,

is given by

the integrals being infinite at the upper limit s = oo, or the lower limit s = s3,

where h = 0 and zs A-ZT is infinite ; and so also

f*\ P**^5? A*-s3)ds _ f'i2 «!-« ¿f

= hH+xnhK, (1 - A)£T- zn AJT;

ra-s1 V(sx-s3)ds ^ Ç s2-s

J s — s3 VS J V(sx — ./q\ J s — s3 VS J v/(sx — s3)VS

= h(H-k'2K) + zn hK, (l-h)(H-k'2K)-zn hK;

»2- s3 A*i~ ss)ds C s - s3 dsrs2-s3Y(sx-s3)ds r

J s-s3 VS - -J(10) J s~h ^s J ^{si — h)^s

= h(K- H) - zn hK, (1 - h\K-H) + zn hK.

Similarly, in the regions

«! > o"> s2>s3> o-> — 00,



rs"° sx—er da /"•- sx—s2V (sx—s3)dcr

(11) L« •(«,-«,) A^T) = J_„, „ sfâ i/(-2)

=f(K'-H')-znfK', (l-f)(K'-H')+znfK';

C "St der 2 rs3-crv'(sx-Si)der

J •(*,-<sM-2) J Sx-cr V(—(12) J /(«i-^Vi-S) J *!-ff •(-S)

=f(H'-h2K)+znfK',(l-f)(H'-¥K)-znfK';

/cr — s3 »io- /*s2 — cr /(st—• s3)«*o-

-/ZF + zn/ZT', (1 -/)¿T - zn/ZT' ;

OV

(14) J„ «i-«- i/(-2) J, ^(si — *3) >/(—2)

-(1-/)(JT-Ä") + «/K"';

, ,2 r*3—°" ̂ («i— s3)do-_ r s2 ~ °" do-

J ¡^ !/(-s) =J VJs^sf) TF^j

(15) = -(i-/)(//'-¿2ít') + zs/zt',

/s2—o- \/(sx—s3)dcr /" s3 — er der

s~P~c ,/(-2) = J •(•1-«8) 7(^2)

= -(l-/)ZT+zs/.ZT';

these integrals (14), (15), (16) being infinite when the upper limit cr = sx, or

the lower limit a = — oo, where /= 0 and zafK' is infinite.

Putting A = 1 or / = 1 in any of these forms will give the corresponding

complete II. E. I., noticing that zn K' and zs K' are zero.

4. The incomplete III. E. I. requires the theta and eta functions of Jacobi,

and Burkhardt has given in his Elliptische Functionen, § 126, a method

of approximating rapidly to the numerical value for given elliptic argument.

But in the circular III. E. I. this argument will be complex of the form

hK + fK'i, where A and / are real fractions, so that great care will be required

with Burkhardt's method in separating the real and imaginary parts of the

theta function.

When /, however, is a rational fraction,

(!) /"l» t' t» !• '"'

a quotient of theta functions, such as



®(hK+fK'i) ®0®(hK+fK'i)(2) ®(hK-fK'i)' ®hK®fK'i

can be expressed algebraically by elbptic functions of hK; a collection of the

simplest results will be found in the Philosophical Transactions (1904),

under the title, The third elliptic integral and the ellipsotomic problem.

5. Make a start now with the complete circular III. E. I.

«,«. lv/(-S) ds2_

s~o- VS'(1) A,B=\

in which the quantities range in the order of magnitude

(2) oo > s > sx > o- > s2 > s > s3 > — oo

and this integral (1) is by Legendre's theorem to be expressed by incomplete

integrals, I. E. I. and II. E. I.

Provided with the notation above we shall find in the sequel

(s) A~£iî=f>fs.iî-/)-K,0fx:

(4) B_ffi^l^.w+KmfK,

(5) A + B = \tt.

To calculate the numerical value we have to determine the co-modular angle 6

and the amplitude angle of eb from the relations (2) and (8), § 2,

,2 «l — «2 • » ■ s. — o"(6) sin2 6 = k'= A-*, sin2 </» = -*

sx — s3 sx ~2

and then from Table IX of Legendre

(7) f=Feb + F(fr),

(8) znfK' = Ecp-fE(\TT),

(9) K=F(\nr,k).

The result in the Weierstrassian notation is given in Schwarz's Formeln,

§§ 59, 60.The relation in (5) is the equivalent of Minchin's equation (21), p. 212,

Philosophical Magazine, February, 1894 ; it is proved immediately by the

substitution



do) (.-... *^k=i)

in A, (1), which leaves ds/\/S unaltered except for sign, but changes the limits

from oo and sx to ss and s2, and thus makes

n^ a r í^(-s) ds

(1) / »i-v«*-*. T|J^'i/ «s s — s3 3

(12) A + B-f ^(-^t(si-sz)(s>-sJ-(s-s^ dsK ' J J [(*l-S3'Ä2-S3)-(*-S3-°"-S3)](<r-S) VS

_i / a — s3sx — ss2 — s

V [(S1-S3)(*2-S3)-(S-S3)(0"-S3)](0"-S)

. _] I Sj — o- • o- — s3 • s — s3

\ [(sl-s3-s2-s3)-(s-s3)(o--s3)](o--s)

6. In the arrangement

(1) oo >s>s,> s2> s > s3> o->-- 00,

there are two remaining forms of the complete circular III. E. I., namely

■7T.

(2) CD- rsiïA-Z) ds .

and we shall find

where in Glaisher's notation

... cnudnu d , a" ,(5) zs u = zn u -|-= zn m + -=- log sn w = -5-log //m,

sn u dît (tu

and also

(6) zc u = zn m + -=- log en m = — zs (A!-— m ),

(7) zdu = znu +-?-logdnu = — zn(K—u).

Thus



(8) -C+D = \-n-,

which can be proved independently in the same manner as (12) § 5.

Hence we can write down the expression of the complete III. E. I. of the

circular form, for any assigned region of s and cr, the result involving a zeta

function to the complementary modulus ; and this zeta function is an algebraical

function when / is a rational fraction ; the formula for the simplest fractions is

given in the Philosophical Transactions (1904).

7. We have seen how A and B change place by the substitution

and the same substitution interchanges C and D.

So also A and C or B and D change place by the substitution

m (-* *=*=!=*)>while the substitution

ro (—„ •■-•■_•;-■)interchanges A and D, or B and C.

8. Two more forms are required to finish the series of the complete III. E. I.,

when it is finite and the parameter is a fraction of the real period, namely,

<2> '-jT.1^ S-*-**■in the arrangement

(3) oo >s>s1>s2>o->s3> — oo,

with

« >-!>£%- '

(Legendre, Fonctions elliptiques, chap. 23 ; Cayley, Elliptic Functions,

§§ 178-186.)9. The proof of his theorem that the complete III. E. I., circular or loga-

rithmic, can be expressed by the I. and IL E. I. is given by Legendre in

Fonctions elliptiques, chapters 23 and 24, and is reproduced in Cayley's Elliptic



Functions, chap. 5. We can abbreviate the demonstration considerably, and

avoid also all introduction of an imaginary in the discussion of the circular form

expressed by A, B C, and D by means of the simple lemma

(i) ,a'iil-«-V*.*Ji=21„.„.,x/ ds s ~ a v 'der s ~ a

proved immediately by performing the differentiation.

Integrating with respect to the elliptic differential elements ds/VS and

der/\/(— 2) beween the limits s3, s2 of s, and cr, sx of er,

r r da ds d \ys r r ds da d |i/(-2)J J V(-Z)YSV*dsa-s J J •#!/(-2)l;<k -•

rHfïvs\« da As pw(-s) ¿s, _,

C Ct \ds da=JJ^-S)vs7(^ry

in which the variables are separated, so that

ds da C C. . ds dap r\ v ds da r r ds da

B = l i ^-^VSV(^)-J J (S"S^7FT)

(3)

y r *- h da p s - s3 tfo

= AT(///' + zn/ZT' ) -fK'(K- H)

= W+ KznfK',

by reason of Legendre's relation

derivable from the relation in (5) § 5, and utilising theorems (13) and (10), § 3.

Taking the limits of s as «3 and s gives a more general theorem

,liVS da fi^-S) ds

X ff_Sv/(_2)+J<a <r-s !/Ä

p <r-s3 da f s-s3 ds

(5) - l~V(sx-s3)V(-ï) JK J,aV(sx-s3)VS

= hK(fH' + zn/ZT' ) -fK' [h(K- H) - zn A.ZT]

= |ttA/+ hK zn fK' -fK' zn hK,


1907] THE ELLIPTIC INTEGRAL IN ELECTEOMAGNETIC THEORY 457

connecting incomplete III. E.I's., as in Legendre's equation (i'), p. 133.

Putting A = 1 gives B as before ; and putting /= 1 leads to a new theorem

(6) X *•*\VS da= Itrh — K zn AA7.

s/(-2)

10. When A is deduced in this manner we must keep clear of s = oo by

writing the lemma (1), § 9, in the form

Sl ~ S$ - S2 — Sia — s-,

in which at

(2) S = oo, lunitfL^-t^-O.W \s-a s-s3J

Integrating between the limits s, oo of s, and s2, a of a, we have

m P*^(-2) <*» C'fWS jVS\ daW J. s-a •Ä+J., V«-t~«-*,/V(-2)

"JJr 3~ »-*, ;•-»•(-2)(in which the variables are separated)

= AJT[(1 -/)iï" - zn/ZT'] - (1 -f)K \_h(K- H) - zn hK]

= JwA(l -/) - hK zn fK' + (1 -f)K' zn AAT,

utibsing the theorems in (13) and (5), § 3.

Putting s = sx, A = 1, gives the result for A,

US A. fio* rfY.r . 3l-S3-«2-«3\ ̂ «*«

and this by the substitution (10), § 5, is equivalent to

(5) A=££y-s)%V(^Ty

while from (2), § 9,



«> A + B = £jy-^%7$Ty

or from (12), § 5, and employing (12), (9), § 3,

C r ds da(8) K.JJ[ff_Aî + iî_8]_ ___

= K(H'- k2K) + K(H- k'2K);

and thus Legendre's relation is proved.

Putting /= 0, a = sx in (3) gives a new theorem,

employing (5), (6), (7), § 2, and (4), § 3, or (5), § 6.

11. For G, integrate lemma (1), § 9, from s to oo and a to s3 in the form

(l)-V(-Z)*t^ + l/Sdf^_^\ s^-s^v ' v 'da s — a ds\s — a s — s3J 3 s — s3

p }•(-£) ds r»(lVS_ WS\ derX S-a Y S Ja \s-a s-sJy(-Ii)

= A/T[zs/ZT'-(l-/)Zr']+(l-/)A"'[A(ZT-//)-znA7T]

= - íttA(1 -/) - (1 -f)K'zn hK+ hKzsfK,

employing (16), (10), § 3 ; from which C is deduced by putting A = 1, s = sx,

(3) C=-\nr(l-f) + KzafK'.

In D we must keep clear of a = sx by writing the lemma

d IV S=s,—s —(4) /(_E)¿(M=S_M=i))_^v ' v ' da\ s — a sx— a J ds s — a l s, — a

and then integrate it between s3 and s, — oo and o-, so that

p/}>/(-2) |>/(-2)\ »is p hfS__doL_X \ s-°" sl-<r )VS X«,s-o-/(-2)



=fK'(hH+ zn AA") - hK[f(K' - H') - zn/ZT']

= \^¥ + fK' zn A-^+ hKznf'K,

employing (8), (11), §3.

Put A = 1, s = s2, and we obtain by means of (8), (9), (10), § 2,

(6) D = i-nf+KznfK' + KJ^f^^ = ^f+KzafK'.\ sx — a-sx — s3

12. To obtain E and Z'7, write the lemma

(i) vz*}j£ + ss£(^-¥-?)-a>-t''a'-a'-* + .tlv ' das— a ds\s — as — s3) s — s3 3

and integrate between s, oo and s3, a; then

X s — aVS X \* —°" s-«3/^S

(2) ^ J *-s3 •Ä J 1/(s1-s3)i/2

»/JT[ A (ZT- 77) - zn hK] - hK[f(K- H) - zn/ZT]

= hKznfK-fKzn hK,

employing (10), § 3, and thus connecting E and F; for A = 1, s = sx, makes

(3) E=KznfK,and /= 1, a = s2, makes

(4) F= Kzn hK+ kA * ~ *' 'S - *» = iTzs AA".\ s — s3 ■ s, — s3

13. With sx and s3 conjugate imaginary, and putting

(1) s1-s2-s3-s2 = ilf2,

^ r YMds , „, f" YMds

(3) K "XtFT)' -^ =X 7R)'and then

l-cn2AA" „ ,„4 sn 2AZTdn 2AZTW s-S2 = Jfr+-^2ÄA-' •s-jft (1 + cn2;yr),.

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/« tu-1 -cn 2/^ ,, „n ,#a4m2/K"dn2/K"

(5) h-'-Xf+xfo • (-*)-*» {1lmy¿y .in which AA" and /ZT' may be replaced occasionally by A and / in the sequel

without confusion.

Employing a formula (Legendre I, p. 257),

tfi\ rl-eoaebdcb 2 sin«¿AoS

W J l + cos^Ä^-l+cos^ + ^ - 2^'

verified immediately by differentiation, and integrating the lemma (1) § 9,

p|j//g ao- r*i(/ —2)j& p Ç" ds da

(7) .jar r^f^dh + hK r\i^K'dfv ' J X 1 + en 2A J0 1 + en 2/ "

■^" (tot* **-»—»)

But when we wish to obtain the complete III. E. I., by putting s = oo,

A = 1, we make both sides infinite.

The infinite part must then be cut out by deducting from both sides

w J„ s — s2l/(-2) » sn2A'

and now

m Ç-(WS jVS\ du i-jv(-?.)d,w X V»-" <-»,M-s) J, • -* •'«

-/•r(=f^r + »^-^—M)

+ «(^^+t^r-/»r—¥).

Here, putting s = oo, A = 1, the first integral vanishes, and

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/sn2/dn2/ \

= K\l + cn2f-m2f)-W

Similarly with/= 1,

H WS da /dn2A \ , _(") Lst^7(^2) = ̂ (^2Ä-zs2AJ-^Ä-

14. To compare B with Legendre's standard form of the ILL E. I.,

(1) f-i , 1 • 2 a^» Av=/(l-^sin2Y),v ' Jo l+nsm2xAX' * v *"

denoted by II (n, k, jg) or IT, when complete, put

(2) s-s3=(s2-s3) sin2 x, s2-s=(s2-«3) cos2 X, V«-(V"-*,) A'X>

¿2 = 32~S3 £'» _ Sl~82

and then

(3) B= fr—'—* f^^^dx =W M sx-s3-a - ss J 1 + n Bin2 x&X '

in which, with Legendre's notation,

... 8, — a, „ <r— s H »P s.—o-

(4) 71 = -^-3, 1 + 71 =-i, 1+ — = -î-,

and Legendre's LT requires the normalising factor i/a.

In the comparison of A, put

(6) S — «,= Ji-2—-, S — S. = (S. — S,)-i-j^, S —S. =(». — S,)-;-^

and then

(7) A= I* — —», fr*'??* ^ = V«[Tl(n',k,x)-FxlKJ \s1 —s3a—s3J 1 + n am2x&Xin which

í-8^ w a~8* 1 . „> 5i ~ °" 1 , ¥ a~82la) n=-, 1 + ra =-, 1-\—, =-,

so that

,., ,„ ,. /. k*\ s, — a-a — s.



the same as before; also

(10) nn=k2, U(n) + U(n') = K+î.

But in all the subsequent calculations it is best to keep clear of the standard

forms of Legendre, Jacobi, or Weierstrass, and to work with the III. E. I.

as the integral of an algebraical function of the standard form in (8) § 1.

For example we find, putting the differential element,

ds

(ii) v~s=

Í19S d*y /?d (/1dy\ qd2y dSdy(12\ dtf = VSds{VSds ) = S d? + *Ts ds'and if we take

/ia, , , ÇVS-VÏ ds<13) lozy=*J-j^rvs'

we obtain the Lamé differential equation of the second order

1 epy(14) -^-2=2s + a-sx-s2-s3,

(15) 4Ï+*£î-(* + '-*-*-*>»-«-Then the function

(16) — ¿i**")

leads to the Lamé differential equation of the second order

ldhz du

(17) l^=^-Ba-sx-s2-s3 + BX2,

provided that

(18) xs — X(Ba-sx-s2-s3)— v/2 = 0,

the spherical pendulum relation.

Landen's Transformation.

15. It will be convenient at this stage to collect together all the formulas of

Landen s transformation of the second order, required in the sequel, adopting the

notation employed previously for the elliptic argument, AA"denoting an argument

to modulus k and fK to modulus k', A and / denoting real fractions, as thereby

the modulus is indicated in the notation and need not be written down.

Without this it will be difficult to reconcile the notation of different writers

on this subject, each adopting a method of his own irrespective of others.



Following Maxwell in his Electricity and Magnetism (E. and M.) § 702,

we select the Landen transformation connecting the new modulus c and c with

k and k' by the relation

, 1-k 2Vk 1-c'w c~i + v c~ï+k, k-i+~c"

k'= ;-,k'c = 2V(kc'), c>k, c <k',1 + c '

all different forms of the modular equation of the second order, connecting

the modular angles 7 and ß shown in figures A and B, such that

< I"8™/3 * 2/1 10S BGcos 7 = c = =——;—ñ = tair ( ±ir — iß) = -¡-¡y,

' l + sm/3 V4 2 ' AC

(2), . a I-C0S7 ,, AC OCk = sin ß = z.-= tan-1 iy = -r-~s = -~-¡-.

1 — cos 7 2 ' AC O A

Then the complete elliptic integral or quarter-period F(%-ir, c) or F(c) is

denoted by F, and F(c') by F', K and K' denoting F(k) and F(k'); and

(3) F=(l + k)K, K=l(l+c')F,

(4) F' = 1(1+ k)K, K' = (l + c')F',

(5s F'-iK' K'-2F'(ö) I-Î' K % F'

16. The geometrical interpretation of Landen's transformation is seen in

fig. B, where we may put

(1) a = am hF, a>' = am ( 1 — A ) F, % = am 2AA",

(2) 6 = 2co = x + X-> 2cù =ir — x + X>so that

(3) x = 27r + w — •'> am 2AZT= \ir + am hF— am(l — A)Z^,

sn2AA"= cos [amAZ^— am(l — A)7^]

W =cnA7Tcn(l-A)7T+snA7Tsn(l-A)7T=(l + c')Sn^T^^>

Landen's first transformation ; and putting A = 0 leads to (3) § 15.

Thence also

_._ l-(l+c')an2hF dn2hF-cen 2hK=-—5—rfs-= 71-ttj—rs

dn hF (l-c)dnhF

(5)dnhF-dn(l-h)F CQ-CQ'

1-c' ~= CM- CB = COSXy



Fía. A.


4651907] THE ELLIPTIC INTEGRAL IN ELECTROMAGNETIC THEORY

the geometrical interpretation, since CQ, CQ', CB = CA dn (/, 1 —f,l)F;

&

(6) dn2hK =l-(l-c')an2hF_ dn2hF+c' = QQ'

" dn AZ^ _(l + c')dnA7^ AB'

Equation (4) can be written

U 1 + c 1 - dn 2hFCO *"* 2hK= Y^d 1 + dn 2A7"and thence

1 -kan22hK(8) dn2hF=T+nratf2hK'



which is Landen's II transformation, in conjunction with

m ,n2hF i1+ *)*****(9) Sn2hF=T+katf2hK>

„A, cn2hKdn2hK(10) cn2Äi"=T+-1^2Ä7T-

These may also be written, replacing 2A by A,

(11) sn hF=\ •(1 + sn 2hKl + k sn 2A7T)-f /(l-sn 2hKl-k sn 2AZT),

(12) cnA7^=Ji/(l + sn2AZr-l-&sn2AZr)+iv/(l-sn2A7r-l + ¿sn2AAT),

dn 2AZT+¿ en 2AZT_1-k

~ 1+k " - dn 2AA- k en 2A7T

(13)\l-k idn2hK+kcn2hK

~ S¡ ï+lc -\j dn 2AZT- k en 2AZT

17. The associated transformation of the complementary period K or F' can

now be written down and interpreted geometrically in fig. A where

(1) f=am2/7", <£=am/7T', cb'=am (l-f)K', 2cb=ir+-^', -ir-2eb'=-^-f',

and as in Landen's II transformation in (11) § 16,

sin cp = ain ^(y¡r + -*¡r')= Ji/(l + sin i/r-l + sin i/r')—1^/(1—sin yfr 1—sin y]r'),

sn fK = I i/(l + sn 2fF'- 1+c'sn 2/7")-i /(l-sn 2/7"- 1-c'sn 2/7")

(2) (1+ c')an fF'

~ 1+ c an2 fF' '

as in (9) § 16 so that /= 0 leads to (4) § 15 ;

cn/7T'=J/(l + sn2/77'-l-c'sn2/77') + Jv/(l-sn2/7" 1 + c'sn 2/T7')

(3) _ cnfF'dnfF'~1 +c'sn2/7"'

, ,_ l-c'sn2/7" dn 2/7" +c'en 2/T7'

dn-^ * l + c'sn2/7" * -T+7-"

(4)w _ 1-c' ll-c' I dn 2/7"+ c'en 2/T7'

= dn 2/7" - c en 2/T7' = AJ lTf7 AJ dn 2/T7'- c'en 2/T7"

and, conversely,

«,„» ,, , anfK'enfK' ¡1 + k |l-dn2/ZT'(5) sn2/7"=(l + ¿) ^dn// ^rr-^r+lnW"



derivable from (2, 3, 4), and the geometrical interpretation of

(6) ifr = \ir + eb — eb', sin yjr = cos (eb — eb'), cos i]r = sin (eb' — eb);

rng^ l-(l+¿)sn2/7T' dn2/7T'-¿en ¿JJ* - dn/ZT' ~ (l-k)dnfK'

COdn/A'- dn (1 -f)K' _ BP - BP

-" \-k ~-RCr~^~BC = GOii'V',

dn2fF'=--{1-k)sn2fK' - dn2fK' + Jcdn fK ~(1 +k)dnfK'

(8)dn/7T'+ dn ( 1 -/) K BP' + BP PP'

1 + k ~~ BC+ BC~~CC' = ca*^'

the geometrical interpretation.

The two circles in figures A and B, linked in perpendicular planes, may be

taken to typify the magnetic and electric circuit, associated in a similar manner.

18. In Legendre's notation, the II. E. I.

(1) E(co,c)= I V ( 1 — c2 sin2 co)dco = I Acûdco,

and the complete II. E. I., or E(\ts-, c) or E(c) is denoted by E, and E(i)

by E', while as before E(\ir, k) is denoted by H, and E(k') by Z7'.

In Jacobi's notation, with a> = am hF, and as employed already in § 3,

(2) f dn2hF-Fdh=EamhF= hE + znhF.

Squaring equation (13) § 16,

, ,, „ 2 dn2 2A7T + 2k en 2A7Tdn 2A7T - k'2(3) dn2 hF=-(TTT)1-,

and then integrating with respect to A, there results

hE + zn hF 2hH+ zn 2hK +kan 2A7T- ¿'2A7T(4) F (l + k)2K

ii- ■,„ 2hH- k"hK +zn2hK+ksn2A7T(5) hE+znhF=-n~1->

J. -f- K

so that, putting h = 1,



and conversely

(8) 77= B+/f, UK' = EF' + c'FF',1 + c

so that, as in Maxwell E. and M. § 701 when corrected,

(9) \(F-E)-cF=2^,

(10) 2(F-E)-c2F=4*~^,

(11) (1 + c'*)E-2c«F = ¿l±n^^.

19. Then7 „ zn2AZT+¿sn2AZT

(1) zn hF-j-^-,

which it is convenient to write

(2) F zn hF = ZTzn 2 AZT + Kk sn 2 A7T,

and conversely

TTzn 2Ä7T = T^zn hF - iFc2*"^^2 dn hF

(3) =i77znA77-J77zn(l-A)77

i et oT-rr nE72sn2A77cnA77

= ^ZD2^-^CT+dn2A77-

Squaring (4) § 17 and integrating with respect to /, we obtain the remain-

ing relations2E'

(4) 77' = rT?-(l-c')7",

(5) 7" = H + k^, E'F=H'K+kKK',

/as mtrr ^2fF'+c'an2fF'(6) zn/ZT =-j-7-^-,

which we write

(7) TTzn /7T' = \F zn 2/T7' + \Fc sn 2/T7'

(8) ZTzn(l -f)K'= - \Fzn 2/T7' + |7Y sn 2/T7',

and conversely

T'zn 2/T7' = 27Tzn/7T' - Kk**^^



= KznfK-Kzn (1 -f)K

(9) JT 9/7T< ^,»2sn2/ZT'cn2/7T'= Kzn2fK-Kk t + dn 2/g»

= 7Tzs2/7T'-7T^,

(10) F 2n ( 1 - 2/) 7" = ZTzs ( 1 - 2/) 7T' - 7T¿ tn 2/7T',

theorems required in the sequel for the quadric transformation of M, the mutual

induction of a helix and a coaxial ring.

20. Integrating (3) § 19 with respect to A,

W log eöTT - 2 log ëcLF + log dn hF'

Q2AZT /®hF\2 ©AT7 e(l-A)7T

') eöF'îi) W ©T* '

the quadric transformation of the theta function.

Similarly by integration of (9), § 19, with respect to/,

e2/7" (®fK'\2 ®fK ®(l-f)K'

(0) ©07" _V0OÄ"'/ ~©07T' " ©A"

The same notation is useful for expressing the change in a theta function

and eta function from imaginary to real argument and comodulus in the form

®(2fK'i,k) H(l-2f)K H(2fK'i,k) .f,H2fK'w ®(Q,k) ~q HK' ' H(K,k) -%q UK' '

where q = exp (—nrK'/K). (Jacobi, Fundamenta nova, or Werke, I, p. 215 ;

Cayley, Elliptic Functions, p. 151.)

These transformations may be used for the bisection of the elliptic function ;

/ ©AT7 \2 ©2AZT dn 2AZT - k en 2ATT(%KF\

W \MF ) ©OZT 1-k

1 @(1-2A)7T ¿ 77(1-2A)7T

r^T* ©OTT 1^1 777T '

/©AT7 \2_1 @(1-2A)7T /fc 77(1-2A)TT(6) VWJ~Ï+1 ©Ö7T 1+-* ^"7777 '



Algebraical form of Landed s transformations.

21. The algebraical equivalent of our first quadric transformation in § 16

is (Tannery and Molk, Fonctions elliptiques, formulas XXII, XXIV)

m m-(.-.,)-l'-W<(W-'.»]',

(3) rn~(s-s3)= ^¡ _ ^ = ^ (< _ ^)2,

so that the graph of a relation ( s, t ) is a hyperbola ; and

(4) 2m V (sx -sf) = V (tx -t3)-Y(tx- t2),

(5) 2m V (s2 -sf) = V («, -tf)-V(tx- tf),

the equivalent of the modular equation (1) § 15, with k, k' in terms of sx,s2, s3,

as in (4) § 2, and c, c the equivalent for tx, t2, t3. Also

,,, m^_( t-tJ-(t1-t2-tx-t3)

W dt~ ' 4(¿-í1)2

co -3^=('"°28~(!-o2¿1"'3)^'where

(8) S=4(s — sx-8 — s2-s — sf), T=4(t — tx-t — t2-t — t3),

W mVS-^VT'

This algebraical result is obtained by substitution in (4), (5), (6), § 15, of

(10) sin2 x = sn2 2hK= -*-3, or -3, oo > s > sx, or s2> s> s3,s ~ ss S2 —' s3

(11) sin2 a> = an2hF = ^-3, or —^-^, oo > t > tx, or t2>t>t3.t —13 . t2 t3

In the associated complementary transformation of § 18, writing a and 2

for s and S, t and T' for t and T, the region Sj>o->s2 is excluded, because

a — sx and a — s2 must have the same sign ; but

(12) sin2 cb = sn2/7T' = S-^^, s3 > a > - oo .


1907] THE ELLIPTIC INTEGRAL IN ELECTROMAGNETIC THEORT 471

(13) sn2/7"-.*J^lI or ^=^, tx>r>t2, or t3>r>-X>,

and these substituted in (2), (3), § 17, will satisfy the algebraical relation above.

On applying this algebraical quadric transformation to our III. E. I.,

(14) wV(-2)=(T-^)28-T(^-)V^~-^/(-7'),

(liï m2(s a) (*-T)[(*-*i)(T-*.)-(<i-V'i-*.)](lb) m(s-a)- 4(í_íi)(T_íj »

2mi/(-2) i/(-7") (ti-f|.<i-<3)1/(_r')

^ ' •-» *-T (t-íJ^Í-^-t-íJ-íí,-«,.«,-«,)]

i/(- 7") T/(-r>) t/(- 7')Í — T "+" M — T T — Í, '

and on putting

(17) t-tx=t>-t2'ti-t3 *- du

u-tx ' vT~ ^Z7'

(18) (t-tx-r-tx)-(tx-t2 tx-t3) = (tx-t2 tx-t3)T^^,

so that

i/(-2) _<fo_ _ •(-7") <ft _ d(-T') du _ V(-T') dt

C ' s-o- i/zS1- í-t ,/7 u — rtfU tx-r VT'

The hyperbolic graph connecting s and t being drawn it shows that as t

increases from tx to oo, m diminishes from oo to tx and s diminishes from oo to

sx and rises again to infinity so that

ON f ^(~2) ds f ^(~r) A /i*-**'*-*s\ i Ah-t*)dt{M)XX —' VS— X «-T •r+VU-T-í.-íJJ " ,/T7 '

(21) 2C(fK) =- 2A(fF') + mf^/F F = -A(fF') + C(fF').

But as t increases from t3 to t2, u diminishes from t2 to l3, and s increases

from s3 to s2 and diminishes again to s3, so that

i-ooï p^(-2) ¿s f'w(-T') dt j/r-t2-T-t3\ rv(t-t3)dt

(22)X *-* VÄ~X, t-r VT + ̂ ytx-T-tx-t3)X ' VT '

(23) 2Z)(/77') = 2B(fF') + C-^ßß^ F= B(fF') + D(fF').



Making use of the expressions for A, B, C, D in § 5, § 6, equations (21),

(23) are equivalent to the single relation

(24) 2K zs/TT' = F zs/T" + F zn/T",

as before in 8 19.

(1)

22. For the transformation from t back to s in the III. E. I., we take

t — tx V(s — s2) + V(s — sf) V(sx — a) — V(s2 — a)

tx—r ^(s — s2) — V(s — sx) V(sx — a) + V(s2 — ay

»on {jZl _ o V(s-s2)V(sx-a)-V(s-sx)V(s2-a)

{) tx-r fr(s-s2)-V(s-sx)-\[V(sx-a)+V(s2-a)]>

[•(«, -a) + V(s2 - a)] [V(sx- s2)V(s - sx)~\

m h-T_X {s - s2)V(s - a) + V(s - SX)V(s2- a)~\

U t-r 2(sx-s2)(s-a)

V(sx — as2-a)- V (s - y s - s2)_i + 2(S-<7)

(4) *^[5P-2 •(*,-. ),

,gy iVi-F) dt rt V(sx-as2-a)-V(s-sx-s-s2)-]V(s3-a)ds

W t-r 7T=U+-

«l. 'a

2(s-<r) J VS

^V(-T') dtt — r VT

K-' r''1V(sx-s3)ds r\V(-ï)ds ÇV(s3-a)ds

-\sx-s3XH¡>i~ VS ~+J s-a VS *J (s-a)V(s-ssy

the last integral being zero as s ranges forward and back again.

According to the region of t,

(7) tx>r>t2, or t2>T> — oo,

and with«3 > a > — oo ,

the relation (6) is equivalent to

(8) A{fF')-K^£- <AfK')>

(9) -B(fF') = K^^,-D(fK'),

(10) G(fF') = 7T^|g; + C7(/7T'),



(11) D(fF') = KC¿j^, + DUF7),

and these reduce to the single relation

(12) FznfF' = KzafK' - K^^..

23. The algebraical equivalent of the second quadric transformation in (8),

(9), (10), § 16, is in a similar way equivalent to a hyperbolic graph between x

and s,

\2>

(2) M2(x-t2)=l8-S3-V^-S3-S*-S^V ' \ 2/ « — «j

(3) Jg(,-Q, fr-l + ̂ V-V*-*)]'

and as in the modular equation (1) § 15,

(4) MV(tx-t3)=V(sx-s3) + V(s2-s3),

(5) MV(tx-t2) = V(sx-s3)-V(82-s3).

Also Mm = 1, and

(6) M-dx _ (* —«,)' —(*i —V«, —«,)ds~ (8-S3)2

(7) MVX = {S- 'a)> ~(8{S__~y3'*2~ Si)V S,

,_. dx ds __. ., .(8) MVX=VS, X=4(x-tx-x-t2-x-ts).

The region t2 > x > t3 is now excluded, because x — t2 and x — £3 must have

the same sign ; but

(9) ain2Ç = snt2hF=t^f, »>»>«,,x— t3

(10) sin2 x = su2 2ATT= ** ~ 3 or "" ' , oo > s > sx or s, > s > s3 ;8 — S3 si — 8s

and these substituted in (8), (9), (10) § 16 will give the algebraical result above.



In the associated complementary transformation with a, 2 for s, S and

x , X' for x, X,S ^~ (T s — s

(11) sin2 cb = sn2 fK' = —- or —-3, Sj>o->s2, or s3>a> — oo ,sx — s2 sx — a

f _ y.' f _ £(12) sin2 y]r = sn2 2/A"' = ~- or j-\, tx>x>t2, or t3>x> — 00,

"\ ¿2 ^1 x

and the algebraical relation is satisfied when these are substituted in (5) § 17.

In the transformation of the III. E. I.,

(13) M2(x-x') = ^^-¡¡f::^^^2-^,

(14) M3V(-X') = {a-S3)î-}_[-^S2-S3)V(-^),

MV(-X') = (s - s3) [(a - sj -(s¡-s¡.s2-s3)]V(- 2')

x-x (a-s3)[(s-s3-a-s3)-(sx-s3-s2-s3)](s-a)

(15)_^(-2) • (-*')

S — a s — a" '

where

(16) ff-S3= *_,,-' ff-S2=—«r^sT"' *-•»- «T-«,-"

In the arrangement

(17) .?! > o- > s2, s, > o-" > s2,

(18) sn2/"ZT = «^ = ^ ^2 = J^g,v ' J sx — s2sx — s2a—s3 dn2 fK

and in the arrangement

(19) S3>(7> — OO, S3>Cr">— OO,

,~~s 1 ,1,-rr, S\~K s, — a cn2fK'(20) sn2 f"K' = -1-% = J-= W4^,v ' J sx — as2 — a dn2fK

so that

(21) /"+/=!•

The integral relation

/22) fW(-x') <fc _ r ¿•(-s) ¿« _ rK(-s") ¿«* ' J cc — x' 1/A7" J s — a V S X s — a" V S'

is now equivalent to

(23) A(2fF') = A(fK')-A(l-f)K', or-Z?(/77')+Z?(l-/)7T',



implying the relation

(24) Z^zn 2/T7' = KznfK' - ZTzn ( 1 -f)K',

as in (9) § 19 ; or

(25) C(2fF') = C(fK' )-C(l -/)7T', or = D(fK ) - D( 1 -/)77',

implying the equivalent to (24), when (5), (7), (8), § 17 are employed :

(26) Fza2fF' = KzafK'-Kza(l-fi)K'.

Thus in transformation I, hF and fiF' become changed to 2AA" and fK', and

in transformation II, 2AZT and fK' are changed into 2AT7 and 2/T7', a succes-

sive application being equivalent to a duplication of AT7 and /T7'.

24. To proceed from s to x by means of the II transformation, take

s — s3 y (x—tf)+V(x—tf) V(x — tf) — V(x — t2)(1) a—s3 V (x — t3)— V (x — t2) V (x — t3) + V (x — t2Y

s-a = V (x - t2.-x -tf)-V(x-t3-x- tf)

K) *-*8 [v(x-tf)-v(x-t2)]\_v(x'-t3) + v(x'-t2)y

a-s3 _ V (x' - t2-x - t3) + V (x - t2-x - t3)

{ó) s-a~ 2(x-x) ~i'

(4) ^P = XV(tx-x),

V (— 2) ds ry(x — t2-x — tf) + V(x — t2x—13) "I V(tx—x')dx

8— a V S~ x — x' VX '

sV(-2) ds flV(-X') dx.„. P''V -2) ds p(6) L -^-vs'i x — x' V X

+P V(tx — x')dx jtx — x r

X (x-x)^(x-*i)~V^"^XV(tx — t3)dx

of which the second integral is \tt, and the third is Z^c' sn 2/T7' or Z^sn 2/T7',

according astx > x > t2 or t3 > x > — oo ;

so that in the region sx > a > s2, it > x > i2,

(7) 2 A (fK) = A(2fF') + i7T- Fc sn 2/T7',

(8) 2Z? (/7T' ) = A (2/T7' ) - Jtt - TV sn 2/T7',

equivalent to

(9) 2ZTzn/ZT' = T'zn 2/T7' + TV sn 2/T",

Trans. Am. Math. Soc. 3«License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


as in (7) § 19; and in the region s3>o-> — oo,i3>x'> — oo,

(10) 2C(fK') = C(2fF') - J»-¿¡r|JF,

(11) 2Z>(/ZT') = (7(2/7") + ítt - sV^F„

equivalent, as in (7) § 19, to

(12) 27Tzs fK = Fza 2/T7' - F ns 2/T7'.

25. Another form of Landen's transformation may be added here ; put

(1) k = th a, k' = sech a, then c = e_2a,

(2) c' = th S, c = sech S, then & = e~2S,

and then, as a form of the modular equation of the second order,

(3) (e2"-l)(e2S-l) = 2, sh2ash2S = l.

Now with co = am AT7, put Aw = th f,

eh« • (ch2 f - ch2 S) shS8inû) = _, C0SÛ) =-_-, cn(l-A)77=sR,

(4)_77 dco ch SdÇFdh = -— =

Ata- V (¿h2 Ç - ch2 8)'

and from (7) § 16,

(5) k sn2 AZT= e-2i, sn AZT = e-i+s.

In the plane xOz of Fig. A, v and w of § 24 are elliptic coordinates, to employ

on Weir's azimuth diagram ; while ' of P' both on latitude X, where

(6) sin\ = th2a, cos\ = sech2a, tan\ = sh2a, tan JX = th a = &.

We can put, for the conformai representation of the stereographic projection,

(7) z + xi = a tan (eb + ai),

x sh 2a z sin 2cb(8)

(9)

a ch 2a + cos 2eb, a ch 2a + cos 2$'

PA, PB rx,r2 e**

a a i/[i(ch2a + cos20)]'

and if OPP2 is a straight Une

(10) ^fc-iTT.



But if APP" is a straight line

(11) yjr + i¿r" = -ir,

and yjr, ■$•' are biangular coordinates of P, so that yjr" = am 2(1 —f)F', with

the value of yjr, ef>, eb' in (1) § 17 ; also

(1-k) AT} a(l + k)

<12) BF = ^nfK' AP = ^VfK>

n«\ np np' «(l-¿) AP- 4P» «(!+*)(13) BP=BP =dn{1_f)]r, AP = AP =dn(1_y)jBr.

(14) OC- OC' = a2= OB2, PB BP'= OB2^1 ~k^ ,

(15) OC' = ®, OC=ah, C'B=a(^-l\, BC=a(l-k).

Coefficient of mutual induction of two coaxial circles.

26. Apply the preceding method to Maxwell's § 701 (E. and M.). The

mutual induction of a short element of wire at P perpendicular to the plane

APB of fig. A and of a circular wire on a diameter AB in the perpendicular

plane of fig. B, is, for the arc A Q where A O Q = 6,

008 € -7 C° a cos ( 7r— ' ) d&<» m^¡.V(a2+ 2aA cos 6 + A2 + b2f

an integral composed of incomplete I and IL E. I.'s : and this must be inte-

grated round the circle A QB to obtain the vector potential G at P perpendicular

to the plane APB ; and then multiplied by 27r^4 to obtain Miov the two circles

in parallel planes.

According to our method of leaving the algebraical expression intact, we first

substitute

(2) PQ2 = a2 + 2aA cos 6 + A2 + b2 = m2(tx - t),

and supposing t = t3 and t2 at A and B, where 8 = 0 and -it ,

(3) PA2=r\=(a + A)2+b2=m2(tx-t3), PB2=r22=(a-A)2+b2=m2(tx-t2),

PQ2-r\ = 2aA(l + cos 6) = m2(t2 - t),

r\ - PQ2 = 2aA(l - cos 0) = m2(t - t3),so that

(5) 4aAcoa8 = m2(t2 + t3-2t), r\ — r\ = 4aA=m2(t2 — t3),



(T) ..„..^^ „_**»*££.V(t2-tt-t3y PQ rx VT

r2"2-rraA coa (ir-6) d6

X ~PQ~(8)

C2' 2-n-aA cos (w — 0)d6 „ fh 2t — L— t. dtM=X -TQ—t—^l TJ^tJVT

dt O — *2-h ÇV^-t^dt= 4-rrr fJ=i-* fcrizif

VT

and then by (10) § 3 and (1), (4) § 2,

(9) JK"=27rr1[2(JP- A)-c27T],

as in Maxwell, with the original sign changed ; and by the quadric transfor-

mation employing (10) § 18

(10) M=8m1^-jr-==4sr(r1 + ri)(K-B') = 8irV(aA) ~ ,

doubling Maxwell's original result, and to be divided by 109 to bring it to

henries.

Maxwell's modular angle 7 is now seen to be AEB in fig. A where AE is

the tangent from A to the circle on the diameter CC, where PC, PC are the

bisectors of the angle APB ; and then BE is at right angles to .4Z?, so that

A, B being limiting points of this circle

ms EB FB r2 '(11) C0S7=_=__ = C.

Putting k = sin ß, so that ß is the modular angle of the period K and K',

AC-CB OO AC(12) sin ß :

rx + r~ AC +CB~ OA~ AC

so that ß is the angle O CD in fig. B where CD is perpendicular to ^1Z?, and

CD the tangent at D.

27. A geometrical interpretation can be given in Landen's manner of these

transformations in the plane A QB of the circular disc or wire in fig. B, as

well as in the perpendicular plane APB of fig. A, in which PC, PC, the

bisectors of the angle APB, are axes of the elliptic cone whose vertex is P and

circular base A QB.

The point F in fig. A where 7* G meets the axis of the circle A QB is the

center of a sphere of which G and P are limiting points, so that P Qj Q C is a

constant ratio.

Now, co denoting the angle .4Z? Q in fig. B,

(1) PQ2 = r\ cos2 co + r\ sin2 co,



and at D, where PDG' is the tangent plane of the cone,

0 OC rx-r2

(2)V T

7-j + r2 rx + r2

4rx r2 a2(3) ABD = am AT7, P7>2 -r.r., CD2 = AC ■ CB = . l 2 vW ' 2 (^l + r2>

so that, round the circle -4 QZ?,

(4)PQ PD rx + r2

QC~DC~ 2a 'and so also

K > QC~ 2a ' QC~rx + r2~

On fig. B the angles QCA, QCA or CQO are denoted by Xi Xi an(*

QBA, Q'BA by co, co'; and we put, as in § 16,

(6) % = am2AA, co = am AT7, co' = am (1 — h)F,

(7) PQ = rxdnhF, CQ = ^^dnhF= CAdnhF,rx + r2

and then, since OQC=x , OQC = x-i sm X = & sm X> cos X = ^n 2 A A",

(8) CQ, CQ' = a cos x ± a¿ cos # = a (dn 2A77± k en 2A7T),

(9) PQ,PQ' = l(r + r2)(coaX' ±keoaX), PQPQ' = rxr2,

(10) QQ' = 2acosx' = 2adn2ATT,

.a sin 8 sin co cos co(H) amx=-öc- = (rx + r2)—liQ—,

(12) (rx + rf)2 cec2 x = rl cec2 w + r\ se°2 w>

COS Y(13) (r, + r2) ¿r^ = rxcotco-r2tanco,

cos v'(14) (r, + r.) —.-= r, cot co + r. tan co,v / \ i 2/ sin % ' 2

(15) sin2 co = \( 1 — cos 8) = \(1 + sin y sin y' — cos x wa x ),

(16) sin co = Jv/(1 + sin Y-l + sin y') —Ji/(l — sin Y-l — sin y'),

etc., the geometrical equivalent of the Landen transformation in § 16.


480 A. G. GREEN HILL : [October

28. In the transformation of M to Maxwell's second form in § 702, we

notice that, from elementary geometrical considerations,

add dx add 2a dX . , . . , .W ~rTñ =-n ~=tT7\ —-;-ti sinY=»fcsinY) cosy = Ay,v ' CQ cosy F Q r, + r2 cos %

2

2iraA cos 8d8 — SiraA f* dx

%A r/kain2

2 ■

Sir a A

,_. 87ra J. r / k sin2 y \ -,

<2) -rTTFjl^F-008^^

87TÍ7./1 /"r /I — A2v \

= r^r2X {—E^-*«»x)*X-*"{ri + r1){K-H).

Hence another method in place of Maxwell's § 702 for drawing the lines of

magnetic force M= constant, employing Weir's Azimuth Diagram covered by

confocal conies for which rx ± r2 is constant.

Denoting Weir's hour angle by a and latitude angle by X, then

r.-r, 2a M K-H(6) sin a = -Lg—-, cos X =-, k = sin a cos X, 3— =-— .v ' 2a r, + rf oira cos X

Supposing the curved line CPC in fig. A to represent for a moment a line

of magnetic force of constant M, starting from C where X = 0, and orthogonal

to the lines of constant fi, such as those shown radiating from A,

(4) k = aina, ~ = K-H.

At C where a =90°,M K-H

(5) ¿ = cosX, g--jr-.

To find an intermediate point F on CPC assume an arbitrary value of k

and K — 77, less than the value at C, and calculate X and a from the relation

M(6) cosX=(7T-Z7)-g7ra,

h M K-H(7) sin a = _ Qx ' cos X 8ira k

We thus require a table of K—H and (K— H)/k, say for every five

degrees of the modular angle 8 = sin-1 k ; the work of redrawing Maxwell's

figure XVIII of the lines of magnetic force for a circular current is being car-

ried out on this method.



The meridians and parallels of latitude of a stereographic projection on fig.

A, with poles at A and B, will map out the electromagnetic field of a straight

current through A and the return through B perpendicular to the plane, by

means of the longitude for magnetic potential, and then J7= 2a = log (rx/rf).

29. Putting cos 8 = c, the algebraical form of the quadric substitution is

a2 sin2 8(1) y ov y-y3 = m2(s-sf)= p(^ = sin2 QPN,

of which the graph is a hyperbola in the coordinates ( c, y ) ;

(2) yl-y = rn2(sx-s) = [a-^C^J,

(3) y2-y = m2(s2-s) = [a^^J,

because

(4) a2c2+2aAyc + (a2+A2 + b2)y-a2 = 0,

(5) (ac + Ay)2=A2(yx-y)(y2-y),

fas (r* + r*\* ( 2a Y , Ayi ri + r* \y*

W y2~\ 2A )-\rx + r2)' °2 ~ a " r,+ *•,"" Vy,'

(8) y»y»"(z)' "i0*-1'

,Qs „_ -y + v(yi-yy2-y)() • (*%)

. „ _ y/(yi-y)[v/(yi-y) + v/(y2-y)]

• (*%)

„ , _ v/(y2-y)[y/(yi-y) + y/(y2-y)]

t/(2/12/2) '

iJ9_y/(yi-y) + v/(2»-2-y) QQ_v(y1-y) + /(y2-y)

« 1/(2/1^2) « ^

PO^7=i/'(yi-y)+ V(y,-y),

„•„ a ^y[^(yi-y)^2^finy)]»in (7 = / / \ *

i/(yiy»)

tan le = WVi + VVtWy_2 Vy2V(yl-y) + Vy1V(y2-y)



Taking logarithmic differentials of tan \8 and sin 8,

etas dd v(yiy*)dy a d8 dy dysinö 2yV(yx — yy2-yY sinö 2y 2V(yx-yy2-y)'

and multiplying by Vy = a ain 8/PQ,

odd _ V(yxyf)dy a cos 8d8 _ dy ydy^(1L) PQ~ VY ' PQ ~2Vy + VT'

(12) Y=4yyx-yy2-y;

and thence the integrals in the algebraical form for determining Maxwell's

second expression for M in terms of K and 77, noticing that y increases from

zero to y2 and back again to zero, as 8 increases from 0 to it .

With the transformation of (4) § 25

M f"2ch28- ch2 Ç dX

-/4-rrV (aA)~J ch2^ V (ch2 Ç - ch2 8)

pch2o-ch2g+l dt

Vö) ~V J ' ch2£ + l V(ch2t;-ch28)

/2Pf e~2C ¿•(ch2r-ch28)1J Lv/(ch2r-ch28)+2o7r e2i+l J06'

of which the second term vanishes.

Components of electromagnetic force of a circular current.

30. We can now express the components of M, and also of the magnetic

potential ii, in terms of E and F, or 77 and K, so as to be able to select the

simplest form. Thus

dM_ p 27raô cos 8d8 _ 2irb p 2t — t2 — t3 dt

~db=X PQf ~~~~m~X3 tx-t VT

2irb r/tx-t3+tx-t2 0\V(tx-t3)dt

(1) =~^J\ t^Tt 2) VT

—"(Li^*-*')—*"£ on+*■>*-wn

by a theorem analogous to those in § 3,

(2) rtf^V{tl~l*)dt- Cdn2(l-h)FFdh = hE-zn(l-h)F.Jts h — * V J- Jo



Or, with

(3) cos 8 = cos ( y + x) = (cos*'+*cos%)(cosx-¿cosx^

k'

FQ = ï(rx+ r2)(coa y' + ¿ cos x) » dB = (cos y' + k cos y) —^;,

dJf B2traAb p cos Y —& cosy' dy

d6 _ (ri + rÂ'Vo cosx' + ¿cos y cosy'

327ra^46 f dy"fT+TfipJ [(I + ^Ícosycosy-^Íco^x + co^y)]—^

(4) l

B2iraAb Cr,* y«s , , ... ,.„-, ¿y=-4 I [(1 + ¿2) cos2 y - (1 — A2)] ——,

(r1+r2)3kk'iJ U ' K K U cosx

167t5 (1+F)77-(1-F)A-

Next

dJ7 p 27ra^l cos 0( ^42 + aA cos 8) ddW ^cL4=Jf + J0 PÇp

^2dJf f 2tt( accost?)2 d8

-M+~Vdb+ X ~ ~^w~~and similarly

dM __ a2dir7 /*27r(a.4cost?)2dc?

<6) alTa=M+b ~db+) PQ3 >

of which the last integral C is given by

n l-rrm Cih+A^l dt

l™ R2**-*«-'»)' p-<, • (*,-<,)&

(7)

«,-<, J vT + J v/(«1-«3)v/7j

= Jw, [(1^,f )2A-4(1 + c'2)77+ 47;] = frrrÊ-M,

by preceding theorems on the I. and IL E. I., so that

tdM , c4 _ 42 dJIZW ^dZ = ^rV2^ + Tdo-'



dM cl „ a2 dM

(9) aHa=^72F+ b W

and the relation

dM tdM , dM „(10) alla+AdA+h-db=M

is satisfied, required in consequence of M being a homogeneous function of

a, A , 6, of the first degree. Otherwise

B2-rra2A2 p (cos y — & cos x')2 dxo = ■f

(11)

(ri + r2)3 Jo k'* cos x'

32mM1 Ç ( 1 + A4) cos2 x' - ( 1 - k2 ) - 2¥ cos x cos x' dx

('i + 'sYJ A2A'4 "cosy/

64?ra2^2 (1 + A4)Z7- (l-k2)K

(rx + r2)3 k2k'*

(12) .¿Z+tf=47r(r1 + r2)|4(2Z7-A'2ZT).Aï

31. If ii denotes the magnetic potential or conical angle at 7* of the circular

ring AQB, the magnetic components are (E. and M., § 703),

dil_1_ dM_ 4ab (l + c'2)E-2c'2F

"' dJ~ torA~ ~db~~^r\ ~cT6r~or

B2ab (l + k2)H-(l-k2)K

~ (rx + r2)3 ¿¿'4

^_ 1 dJf _-a/+c'_ 1 dM 4a2 7; 1(1+c'2)A-2c277

W d&=~2^rZâ\I'"~- 27T.42 ~2-irb^b~~rx~ 7i + rx~ ~cr2~

or

32a2 277- k'2K 3 (1 + A2)77-(1 - A2)7T

~ (ri + rj V* + r, + rt ~ A'4

Also, from the homogeneity of ÎÎ, of zero dimension,

dii .da .dû h tw ^ 4a2bE(3) a'da=-AdA-b^J = ^rA^M+C^-rf7or

32a2 o 2H-k'2K



In the neighborhood of the axis, where A, rx — r2, c, and k are small.,

(4) F-**(l+$ + %-)t A=^l^-ff..),

(5) T7-A=^(ic2 + T3gc4...), A-c'2T7=l7r(lc2 + l|c4...),

(6) (l + c'2)A-2c'2T7=^|4,

,vs 2H-k'2K /. 9A2 \

(7) —¡r- =*H1 + "4-)'

d^- 67r2«2^-26 102ir2a2A2b

{) ~db~ r] ' °r ~ (rx + rj '

/on *,r ^ i 16a2 ^2n /, 3c2 \ 47r2aM2/ BaA\

(9) M+C-Wi—t-**\}+T'~)-,x—(I + -7r-Jor

, . . 16aM2 , ,„ o72N B2ir2a2A2\, BQa2 A2 "I= 47r(r1 + 7-2)7-■-r4i-n-(l + £A2) = ;-•-^ l + ?-■-^ ,

dii_37ra2^4ö 967raM&( ^ 31 ~ rj ' °r (7-J + 7-,)5'

dß_ W* oira3A IS-ira2 V Boar A2 "1 967ra2^42

w w ,* +-if-' (vt^?L +(vMtfJ~ ta+*■,)■On the axis itself, .4 = 0,

díi A dß 27ra2

(12) dz=0' do-=-7.r-

A simpler resolution of the magnetic force is into components Gx and G2,

perpendicular to PA and PB, and we find (A. Russell, Philosophical

Magazine, April, 1907),

and on the axis A = 0, Gx= G2 = ira/r\, but (9, and (r2 are infinite along

the wire.

A similar resolution of the gravitational force of the potentialJad8/P Q of the

circular ring AB will show that it can be resolved into the components 2 Gx

and 2 G2, but now in the direction of PA and PB.



Galvanometer constant of a circular coil.

32. To calculate the galvanometer constant G at a point on the axis of a

coil of wire of n turns in the form of a ring of given cross section a, we have

to evaluate the integral

7i T Ç2irx2dxdy 2-¡rn Ç ydx(1) (x=aJ J (x2+y2)i = ~a'X (x2 + y)»'

taken over the area or round the perimeter.

For a circular section AQB, with the axis of the ring through C in fig. B,

2-n-n p a2 sin2 8d8 _ p 2n sin2 8d0(2) G==™2X C'Q "J, CQ '

and putting as before in § 26, with OC = a/A,

(3) CQ2 = m2(tx-t), C'Q2- CB2 = ~(1 + coa 8) = m2(t2-t),

C'A2- C Q2 =2^-(l - coa 8) = m2(t - t3),

we have

(4) m2(tx-tf)=CA2=a2(±+l), m2(tx-t2) = CB2=a2(±-iy

(S\ c'~ ¡tl~t2 1~k

2nAW /"*. „ dt

-J-l^-^-^VT

2nk2""3r-f [~2 v7 ̂ v/ 7-+ (tx-t3+ tx-t2)(tx-t)-2(tx-t2 ■ t ,-g] ^

~~ 3a4

(6)2wA2w*

= -gär- [(«i - <3 + «i - *,)•(*! - t3)E- 2(tx - t2)V(tx - t3)F]

32ti (l + c'2)E-2c'2F

~SAC c*

The reduction can also be carried out with the functions K and Z7, by means

of the angles x an(I x'> ano-



nm a i ad0 7 dX(7) C Q = lcoaX +«cosx, Wç=k-i^,

ink p , dx 8ti, (1 + A2)77-(1- A2)ZT4«** p71"

(8) G = — j sin2(x + x')cosx'-3 0C7' A2

and (r is given by the same expression when C is moved to any point on the

axis of the tore, the modular angle ß of AT and Z7 being half the angle between

the tangents from C to the circular section of the tore.

33. The form of the result shows that the integral for G and dM/db is the

same essentially ; for the comparison of these and other integrals it is convenient

at this stage to have the following integrals to quote in the sequel :

2" odd 4aF 8aK

(1) I PQ^^-rf+rf

the potential of the ring AB ; and as in § 31,

KJ J PQ3 rxJHtx-t VT rf ' rx

Ç2aA(l- cos 8)d8 4 ft - t3V(tx- t3)dt 4 E - c'2F G2(d) J " ~^Q~ ~-r\J t^rr~VT ~T~ c'2 ~ r,'

Since

d 2a.4 sin 8 _ 2a J. cos 8 2a2 A2 sin2 8

d8 PQ~~ * PQ + PQ1

us T-W-Hrl + rl) (r2x-PQ2)(PQf-r\)K > ~ PQ + 2PQ3

r2r2= IPO — I ' 2

(5) £'pQdd =f^ d8 = 4rxE=2(rx + r2)(2H- k'2K),

reoaddd _42E-(l+c'2)F _ 8 ZT-77

PQ rx c2 rx + r2 k

= iPQf-(r2x + r22)PQ + ^í,

Also



B £ PQfdd=2(t-2x + r22) j PQdS-rlrlj-^L

(8) =87-1(7-2+7.|)A-4r17-277

= 47-J[2(l + c'2)A-c'27']

(9) 3/

= (rx + r2)3 [4(1 + A2) H- JÉ" (5 + 3A2)7T],

ain'fldfl 16 (l + c'2)E-2c'2F 8 (1+A2)77-(1-A2)ZT

PQ ~TX "c4^ ™ rl~+72 ' ~1F~ '

so that as is evident from (7) these last two integrals (9) and (10) are essentially

the same as well as the integral (1) § 30 ; and this shows that G is also the

potential of the tore at any point C" of its axis, the tore being homogeneous

and of mass 2-rrn.

Potential of an elliptic disc, gravitational and magnetic.

34. Begin now with the potential Iôf an elliptic disc

x2 v2 %2

W a2+¥ + ^=1

of uniform surface density 1 ; then (Cayley, Proceedings London

Mathematical Society, vol. 6, p. 42)

_ _ T f Jà x2 y2 z2\ de(2) V=2ab \ V\l--2-,-jan-\-77im-0-¡-\.w A \ a+e b-+e e J V (a2 + e • b2 + e- e)'

where X is the positive root of

X 77 Z

(3) 1-a^+~e~oT+~e~~e=0,

p and v denoting the other two.

Next by differentiation of V with respect to z the magnetic potential Í1 is

obtained of the elliptic disc, magnetized normally with uniform unit intensity,

as represented by the component attraction of the disc perpendicular to the

plane, or by the solid or conical angle subtended at a point ; so that

_ dV /»» 2«6»de

reducing to

/*» Zabzde



p 4abzde r° 4 V (Xpv) de

() ° = X eV(4:-e-X-e-p-e-V) = X ' ~*~ ~ VE '

E=4e — Xe — pe — v.

This is proved by determining x, y, z in terms of X, pu, v, the elliptic coor-

dinates, such that (x, y, z) is a point of intersection of the three confocals of

(3) to the elliptic disc given in (1), when

(6) e = X,p,v,

(7) oo>X>0>/i> — b2>v>— a2>— oo,

and then

a2+Xa2+pa2+v . b2+Xb2+pb2+v 0 + X- 0 + pO + v

(°) x — a2_Q2. „2_o » y —

(9) 1

a2_62a2-0 ' y~ b2-a2-b2-0 ' «r_0-o2-0 '

7/2 a2 e — Xe — /a • 6 — v

a2 + e b2+ e e a2 + e o2 + ee '

Interpreted geometrically, ii is the solid or conical angle subtended by the

elliptic disc (1), or the apparent area, as it may be called ; and Schwarz has

shown that the apparent area or Scheinbare Grosse of the ellipsoid or of any

elliptic section of the tangent cone is given by an expression equally simple

(Göttinger Nachrichten, 1885),

(10) n= r2V(-4-X X.x p-X0-v)de27r_ pV J X (e-X0)V(4-e-X-e-p-e-v) J„

( E0 = 4 ■ \ — X ■ \ — fi ■ A0 — v ),where the ellipsoid is defined by

*2/(-A0) de

X0-e VE

y2

(11) o-Tx/^x/xr1'

reducing to the elliptic disc as its focal elbpse when X0 = 0.

Put

(12) e-X= M2(s-sx), e-p=M2(s-s2), e-v = M2(s-s3)

ande(or e-X0) = Jf2(s-o-),

(13) E=4e-Xe-p-e-v = M6S,

(14) 4a2b2z2 = 4Xpv=- A0=-i/62;

and then, as in (3), § 5,

(15) a=r2-^^=4A = 2TT-4B.K ' J!x S-a VS



35. Schwarz's theorem is proved by considering the equation of the tan-

gent cone from (x, y, z) to the ellipsoid defined by e = X0, which, when

referred to its principal axes, the normals of the three confocals through

(x, y, z) defined by e = X, p, v becomes (Salmon, Solid Geometry, § 173),

K) *-\P r*-\^»-\

one of a family of cones confocal for different values of X0.

The conical angle O of this tangent cone, and the perimeter <í> of the sphero-

conic on unit sphere of the reciprocal cone

(2) {X-X0)x2 + (p-XQ)y2+(v-X0)z2 = 0

ave connected by the relation

(3) a + = 27r.

On the reciprocal cone replacing x, y, z by direction cosines I, m, n, we

may take

w '- *("-') (xh- •-=*)■

^-^(vèr,-7---i:J-

, f ß — v v — X X — p\(5) l2 + m2 + n2 = k (~-^- +-+--)-li

0

subject to the condition

(6) h_*-V»-V»-\.v ' p—v-v — X-X — p

and now

-E0 de2(7) (depy = dl2 + dm2 + dn2 = juzrlj £ ,

(8) E=4e — Xe — pe—v, E0 = 4-XQ — XXQ - p.-X0 — v.

As a point travels round a quadrant of the sphero-conic, e decreases from p,

to i», and

(9) X> X0> 0 >p>e> v,

di, »•"--«f^în accordance with (5) § 5, ii and i> being 4^4 and 4Z?, Schwarz's result.



36. In the notation of Gauss (Cayley, Proceedings of the London

Mathematical Society, vol. 6; Halphen, Fonctions Elliptiques, vol. 2,

chap. 8)

(1) G, G", G' = X, -p, -v,

and in the region /i > e > i» we can put

(2) G+ G'coa2T+ G'ain2 T=X-e,

(3) G cos2 T + G" sin2 T=X0-e,

(4) (G - G") cos2 T=p-e, (G - G") sin2 T= e - v,

™ de(5) dT = —J-.-r,v ' V (4- p — e-e—v)

p dT _ í" de

( } X V(G+ G coa2 T+ G" sin2 T) ~ X„7Â'

(7) UV ( X - v) = liK, T= am AZT.

In the region oo > e > X > X0, we should take

(8) sin2 T= ——, cos2 7= ^—, 7= am hK,v/ e — i» e — v

(9) G + G coa2 T+ G" sin2 T= x~ve->1v ' e — i»

(10) <?'cos2 7+ G'ain2 7= (X° ~ "'* ~ v\ ~_ <X ~ "^ ~ "\

equivalent to the substitution

37. The inequalities

(1) oo>e>X>X0>0>/x>-62>i»>-a2,

(2) oo > s > sx > a- > s2 > s > s3 > — oo ,

show that the form A (fK ) or B(fK' ) is required ; and from (3) § 5,

(3) ii = 27r(l-/)-47Tzn/A',

equivalent to Nagaoka's result for a circle expressed by Weierstrass functions;

and here, as in (6), (7), (8), (9) § 5,

Trans. Am. Math. Soc. 33License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


(4) zn fK = ZfK' = Eeb- fE,

(5) fK' = F(eb,k'), cb = amfK',

so that Legendre's Table IX can be employed for a numerical application,

(6) sin^=sn2/A'=^ = ^-,

(7) C0S^=cnV^=^=_^,

(8) A2* = dnVZT=^=^,Ji "3

(9) ¿2 = S!^S3='^* 7^_8l-S2_X-r*

* ' s, — s,X—v, s, — s„ X — v 'ul "3 ' "1 3

and the modular angle is then half the angle between the generating lines of the

confocal hyperboloid of one sheet, or the focal lines of the tangent cone.

38. Thus for instance, if we takef= J,

^=x"^-*-VM-7- l + l-l-O- k-^f-W

p = — kX, v =1-A'

and now

(2) zn \K'= \(l-k), a = ir-2(l-k)K.

At an infinite distance,

(3) X=oo, ûnep = l, A = 0, K=\ir, A'=l, K'= oo, A"=l,

(4) znfK' = Ecb-fE'=aincb-f=l-f,so that

(5) n = 27T(l-/)-27r(l-/) = 0,

a verification.

Inside the focal ellipse, in its plane,

(6) X = 0, f=0, îî = 2tt;

but in the plane outside the ellipse

(7) p = 0, f=l, znA' = 0, Í1-0,

to serve as a verification.



Along the edge of the focal ellipse

(8) X = 0, p = 0, A'=0, zn/A' = 0,

(9) il = 2,r(l-/),

and two surfaces of constant ax and ii2 intersect along the edge at an angle

(io) H«2-"i) = ^(/i-/2);

see Maxwell, E. and M., § 487.

Along the focal hyperbola

(11) p = v=-b2, k = 0, A=ítt,

A' = l, 7" = 1, Eeb = aincb,

zn/A' = sin cb—f,so that

(12) a = 2-K(l-aineb) = 2-jril-J^-\,

and the conical angle is now bounded by a cone of revolution of vertical angle

7r—2eb (Cayley, Proceedings of the London Mathematical Society,

vol. 6).

On the axis perpendicular to the plane of the ellipse

(13) as —0, y = 0, z2 = X, b2 + p = 0, a2 + i»=0,

p "2-*2 k* V + *2a2+Z2' ~a2+z2'

dn2(l-/)A'=l-^2=e2.

39. The equipotential surface ß = 2-irf, a constant, cuts the plane of the

ellipse (or any other area whatever) along its edge at a constant angle irf; this

is evident from simple geometrical considerations of the conical angle or appar-

ent area of the ellipse as seen from a point close to the edge, when the apparent

area on the unit sphere is cut out by two diametral planes, one parallel to the

plane of the area, and the other drawn through the adjacent tangent of the area.

Magnetic interpretation of the potential of the elliptic disc.

40. In the potential of an elliptic disc, as given by Cayley,

T-^ru-i-á.-jL.)z2 y2 x2 \ de

VE'



when modified by (9) § 34, and employing (12) § 34, with

(2) e=m2(s-a), b2 + e = m2(s - a'), a2 + e = m2(s - a"),

(3) oo > s > sx > a > s2 > a' > s3 > a" > — oo ,

X= 77i2(s1—cr), 62+X= m2(sx— a'), a2+ X = m2(sx— a"),

(4) p=—m2(a — s2), b2+p = — m2(s2—d), a2+p=m2(s2—a"),

v = — m2(a — sx), b2+v = — m2(d—s3), a2+ v = m2(s3— a"),

(5) a2 b2 z2 = Xpv=- im6 2,

(6) (a2-o2)&V = o2+X-&2 + /i,-o2 + I»= |m62',

(7) (a2 - b2)a2x2 = a2 + X- a2 + p- a2 + v = - |w62",

and the terms in V are

de 4dbK. , H de 4abK(8) *ab X vË = vj^vy

(9)4abz2f4w = 4zr^(-^J eVE J^ s-a vs

= 4zA(fK) = 4z\_\-rr(l - f) - Azn/A'],

^ J (o2 + e)v/A-^(a2-62)J s-<x' t/ä

(10)

4aAa;:Ç de 4bx r$V(-2") ds

J (a2 + e)1/7i'-r/(a2-&2)J s - a" VS

4bx

(H) =7(d^W)CV"K')

4 A'/»

= 1/(o»-y) [*-/'*' - Ml -/")]•

Here, as in (3) § 2,

(12)/=f "£/&)' sn2^'=x^' °*fx-Bt;• *■>>*'->£•

(13) /'= f+ r * sn2/'A=^^, cnV'Z^62^, dn2/'A=^,

<I4> r-C+ù£rt -^-w -"-'-to- *"■'"q +M

a2+X"



We have seen already in (5) and (15) § 34, that

(15) il = --J=4A(fK') = 2?r(l-f)-4KznfK',

is the magnetic potential of the elliptic disc with uniform normal magnetization

parallel to Oz, or of the unit current round the elliptic ring; so also

(16) a'- -%= VW-W)EV'K) = 7(£v)K™f'K'

is the potential for uniform magnetization parallel to Oy the minor axis ; and

dV 4b 4b

(1T) °"= - -dx = 7&=V) C{f"K'] = TW-*) [*-/*•-Wi-AI.

for uniform magnetization parallel to Ox the major axis : and íí', il" are the

components F, G of the vector potential of the elliptic current.

For uniform magnetization of the elliptic disc with direction cosines I, m, n,

the magnetic potential is

(18) rail + mil' + IÙ".

Induced magnetism of a hollow ellipsoid of soft iron.

41. We may cite here the expression of the induction of a uniform magnetic

field on a hollow ellipsoid of soft iron, bounded by confocals, of the family

x2 v2 z2

W o?+~x + ¥+~x + x=1,

X. and Xj defining the outer and inner surface of the shell.

We require the three incomplete II. E. l.'s

(2) A,B, G=£de

(a2 + e,b2 + e,e)VP'

1P=4a2+eb2+e-e, A + B+C =

V(a2+ x-v + x-xy

and with the fraction A defined as in § 2 by

C f de



we find from § 3,

W A = aJa^W) ÍKK-H) - zn ÄA] ,

(5) B = b2{a2a_b2)[h(H-k'2K)-zn(l - h)K-],

(6) C= ¿ [-Ä77+zs(l-Ä)A],

(7) A2 = l-~, k' = h.v ' dr a

If the external inducing magnetic field has a potential

(8) V=Xx+Yy+Zz,

we can examine the components one at a time, so that if the component Xx

induces a magnetic potential il in the iron of the shell, il0 in the outside space,

and ax in the cavity, the magnetic conditions are satisfied by putting

(9) ax = Lxx, a0 = LA

°A X,

(10) ß = Lé~^fx + L* 2—r"«Ax— Jí0 Jix— Jia

WllGl*6

Ax-A (hx-h)(K-H)-(znhxK-znhK)

( > Ax-A0~ (hx - h0)(K- 77) - (zn A, A-zn A„ZT)' " ';

and then determining La and Lx from the condition, with magnetic permea-

bility p,

(12) p^-v(Xx + a) = ^(Xx + ax)ov ~(Xx + a0)

in crossing the inner or outer surface, noticing that

d.4 x 1 dX 2px 1 21(13) x

dv a2 + X VP dv a2 + X VP VP'

Other cases for components of V such as xy, xyz, • • • are given in the

Journal de Physique (1881); the hydrodynamical analogues can be deveL

oped by means of the same analysis.



Potential of a circular disc, gravitational and magnetic.

42. For a circular disc put a2 = b2,

W A \\ a2 + e ey(a2 + €)i/e

ellipsoidal coordinates are now unsuitable, and the elliptic coordinates, v and to,

must be employed, with the substitution

(2) a2 + e = a2 ch2 u, e = a2 sh2 u,

(3) x2 + y2 = a2 ch2 e cos2 to, z2 = a2 sh2 tj sin2 to,

(4) rx, r2 = a(ch v± cos w),

rx, r2 denoting the focal distances of P from the foci A, B in fig. A ; and

x2 + y2 z2 ch2 v cos2 w sh2 v sin2 »/>1_L_£._= 1 —_

a2 + e e ch2 m sh2 m

( ch2 u — ch2 v ) ( ch2 m — cos2 w )

ch2 u sh2 w '

r" V ( ch2 m — ch2 v ■ ch2 m — cos2 w ) ,W F=4al-cTOi^-~du>

(5)

dF_ /•» 2a2zde

dz

(7)

=4J" sh v sin wdw

sh w i/( ch2 u — ch2 « • ch2 u — cos2 w ) '

Next substitute

(8) sh2 u = m2(s — a), ch2 u = m2\s — sf),

(9) 77î2(<7-s,) = 1, ch2M = —^, sh2w = ——,°"~V °" — S3

(10) ch2 m — ch2 v =-', ch2 m — cos2 w =-2,

(11) ch2 v = S^"~3, sh2 v = ^- ,

(12) cos2 w = Û3, sin2 to = —~2,

(13) if.4Ui.5J2?, *.ü=ü,V ' Sl — S3 Cn C ri + r2

as in Maxwell's second expression for M, § 701.



The inequality sequence runs

(14) 00>S>S1>cr>S2>S3> — 00,

so that, from § 2, dropping A',

(15) A'-5-/'è(1r/)' cosW = dn(l-/).

If then we write A for V (x2 + y2), x = A cos cb, y = A sin cp,

A , dn(l-/) A dn2(l-/-)

» A' sn /"en f A'(17) - = ahvanw = ~I¿j^ = 1ran(l-f)en(l-f),

(18) r^2=ch7j±cosM) = ^-|±l)dn(l-/).

Sii

(19)

Sincesh udu V (a — s3)ds

■/ (ch2 u — ch2t< • ch2 u — cos2 w>) i/ # '

equation (7), as before in (15) § 34, becomes transformed to

(20) Q = 4£ ^~^ ~ = 4A(fiK) = 2^(1 -/) - 4Azn/A'.

To make s oscillate between s2 and s3, substitute

/oí \ i2 a — h i_2 a — s a — s a — s3(21) ch2 m =-, sh2 u =-, ch2 v =-3, cos2 w =--3.

S ~ S3 S ~ S3 S2 — S3 S2 ~ S3

43. The potential for uniform magnetization of the circular disc parallel to

AB is, as before in (16), (17), § 40,

dV r 2a2AdedV r°

c+«W(1-?t.-ï)X4 ch u cos it> sh m dn

ch2 m y' (ch2 m — ch2 v • ch2 tí — cos2 w)

4 p s2 - s3 •(«, -s3)ds_1K-H^

COS W JSx 8 — S3 V S COS 7/1


THE ELLIPTIC INTEGRAL IN ELECTROMAGNETIC THEORY 499

From Maxwell's second expression in § 701,

(2) J!f=47r(r1 + 7-2)(A'-77) = 87racht)(A-77) = 27rîl',

and this is evident when we notice in figure B that M is composed of elements

,. cos 8 cos 8(3) -pgi, add — pQ add,

which is the magnetic potential at 7* of a strip QQ" parallel to AB, magnetized

longitudinally; and these elements are integrated over the circle AQB, and

multiplied afterwards by 27T.4. to obtain Miov the two circles.

Keplace A by x, then at any point (x, 0, z)

(4) M= 2-wxa' = — 2-rrx -=-, 2ttîî = — 2tt -y-,

da „ d2v i dM(5) 27T-,- = - 2 IT

dx dxdz x dz

(6) 2tt— = - 2tt --= 2tt (-- + - dV\ - - -* ' dz dz2 \ dx2 x dx J x

and a magnetic line of force along which M is constant is the orthogonal

trajectory of the equipotential surfaces il = constant ; in fact M is the Stokes

function of the magnetic potential, and

eFa d2« 1 d_n_

^ ~dz2+^dxY+x~dx~- '

d2M d2M ldM_

( - W + Ik? ~ x ~dx - 'or as they may be written

... d / dil \ d / dil \ .(9) dAX^j+dx(a:^)=0'

d / 1 dJ7\ d /ldM\_

^ dz\x dz J dx\xdx)~~'

These are Maxwell's results of § 703, obtained from independent physical

considerations when x and z are replaced by his a and b ; the symmetry round

the axis z shows that the integration employed in Maxwell's treatment can be

effected, so that his result, where V and il are equivalent, may be written

dM r2" dV ja „ dVa -=— d8 = 2ira

dM (*■

dy dy '

nv\ dM C dV' M o dV<12) -db=-J a~drde = -27ra-df


500 A. G. GREENHILL : [October

The potential of the circular disc is now written, either from (1) or (6), § 42,

P ft °k2 v cos2 w s^2 v sm2 w\ s^ udu

X \ ch2« sh2« ) V [(ch2w—ch2 •>.>)• (eh w—cos2«))]

V(sx -s3s2 — sf) T/(sx — aa — sf)-\V(a — 83)ds

(13)

= 4 C\a AV(8i-sz'sz-s¿ „■>/(si-q"<r-g2)"|1

X, L s-*3 s-°" ] vs

4aK . nt= -r-^lil — zii,

ChT)

of which each term can be interpreted by means of the homogeneity relation

Tr dV dV dV(14) Vwma +A +z.

da^ dA^ dz'since

C"> Í?—Û, í?—*and therefore

(16)

d» ~ ' d¿

dF 4A^ 8aA

da ch» r, + 7*2 '

the potential at P of the circular ring round the rim of the disc.

Similarly in the case of the elliptic disc in § 40 the term in (8)

, , r de 4abKW *abjvË = v(^V)

. may be interpreted as the potential at P of a ring round the ellipse cut off by

a consecutive confocal ellipse, so that the thickness is inversely proportional to

the perpendicular from the center on the tangent.

7Ae Stokes function of a circular plate and ring.

44. Denote the Stokes function of the plate by A7", using b again instead of z ;

then, as in §§ 31, 43,

dlV dV(1) ^ = 27r^5Z = -27rîl'=-47r(7-1 + 7-2)(A-77),

d^=-2-rrAdZ=2-rrAadA db

(2) =2-rrA[27r(l-f)-4KznfK']

= 2irA-4A(fK'),License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


and by analogy

dlV(3) ^ = 27ra-4¿(l -/) K = 2ira [>/- 4Azn(l -f)K"\,

a change of / into 1 —/, and A into a ; and this term is the Stokes function of

the circular ring, of which the potential function is given in (1), § 33 : so that

the Stokes function at P of the circle A QB is 27ra times the apparent area at P'.

Then by the homogeneity of the second degree in a, A, b, of N

dN ÁdN TdN2JST=a~r +AT-r+b-li-

U) da dA db

= 8-rra2A (l-f)K' + Sir A2 ■ AfK' - 4-n-b ( rx + r2 ) ( K- 77),

(5) A7"=47r(a2-^2)Z?(/A') + 27r2^2-27ro(rl+7.2)(ZT-77+7rdn2/A^').

For points on the plate, /"= 0, N= 2-rr2A2: while /= 1 for points in its

plane outside, and N= 2ir2a2; and N— 2-ir2 A2 may be taken as the Stokes

function at P of this plate and a coaxial equal and opposite plate at the level

of P.

Changing the superficial density from unity to l/2irp, we shall see in the

sequel that2?r2^2 — N

<6) —%fp— = M-

where M is the function in § 54 investigated by Viriamu Jones for a helical cur-

rent, or the equivalent cylindrical current sheet ; and the potential of the two

end plates is the magnetic potential of the cylinder magnetized longitudinally,

while the Stokes function N— 2-rr2 A2 is the magnetic potential for magnetiza-

tion across the axis of the cylinder.

The various dissections of an integral.

45. To arrive geometrically at the expression of the conical angle il by means

of the angles 8, cb, i|r, x-, eo defined previously on fig. A and B, we employ

the idea given by Maxwell in E. and M. § 418, that the area il, cut out on

unit sphere by a cone whose vertex is at the centre, is equal to 27r minus the

length <ï> of the curve traced out on the sphere by the reciprocal cone ; this is

a direct result and generalisation of the theorem that the sum of the area of a

spherical triangle and the perimeter of the polar triangle is 27r.

The same idea is employed by Schwarz in his memoir Die Scheinbare

Grosse des Ellipsoïdes, the apparent area and conical angle being the same

thing, either of the ellipsoid, or of the tangent cone, or any elliptic section of it.

If the tangent plane PQY of the cone, shown in elevation and plan in figures

A and 3, turns through an angle d<E> while O Q turns through the small angle



dd, and if the normal line Pq of the cone cuts the plane AQB in q on Y M

produced (fig. B), then Q and q describe curves which are polar reciprocal with

respect to M; so that QMy is perpendicular to qy the tangent at q ; and the

sector velocity of q round M

dd dct>(1) %Mq2 ̂ = ïPq2Tt coa MyP,

dí> _M£Py _ PM2 PQ _ 1 PM

(2) 'dß~PfIfy = PY2 ' PM = ' ~QY2 PQ 'PQ2

so that, in Maxwell's notation, with OM= A,

r2" 1 bdd(3) ftr-O-*- Ç --¿Wâpç.Jo 1 PQ2

Now from § 27

^42 sin2 8 4a2A2 sin2 8 f J2A \2 ■ 2 °de 2h dX

(4) -pQ~ " (r. + rJCQ2 - \rf+Tr2) Sm *' PC " r, + r2 AX'

so that, in Legendre's form of § 14,

(5) *.f *!*,-&K ' X l + wsm2x Ax

( 2A \2 r —r ( A2\ / 26 \2(6) »._(_ff_) jfe.D-1», a=(l + n)(l+- ) = ( —,— ) .KJ \r1+r2J' rx + rf v T ^\ »/ Vi+'V

To reduce <ï> to our form P in § 5, we put

. , s — s, dv /(s. — s,)ds(7) sin2y= --?, -^ = v ' 3I ,v ' K s2-sf AX VS

and interpreted in fig. A, where OM= A, OC= ak, OC = a/A,

W 8^-„-^ 2^ ; ' s1-T3-dn-/ii "V 2J. ; -A 4^2 -kA~OM'

a-s2 _ (rx + r)2-4A2 s_x -_ff _ 4^2 - (rx - r2)2

8.-8."' 4A2 ' s,-ss~ 4A2

¥

A'-£.3

so that, as before in § 34,

a« *=4.C^Ä=4*(/->.X*i X W (_V ^ (7s



46. If da denotes an element of area at any point Q inside the circle, due to

any mode of dissection

C da „ r bda

and the integrals which arise in the determination of these potential functions

change in appearance with the method of dissection employed by the use of each

system of variables ; so it will save repetition in the sequel if we cite here some

preliminary lemmas.

I. The potential at a point F on its axis of a flat ring-shaped disc, bounded

internally and externally by concentric circles on diameter ab and AB, is

(2) UFZ-¡-F-) = 2,(FA-Fa),

and thence also the same for any fraction of the ring cut out by two radii at an

angle 8, replacing 2tt.

II. The potential at P of the line .4P in fig. A is

„ r dx , PA + PB + AB n , . AB „ , PA-PB(3) LFQ = logPÄ+PB--rA-B = 2th PÄ+-pB = 2th -2Ö3T'

and when P is at F, the potential of .4P is twice the potential of O A, and

OA(4) 2th-'FÄ-

In the elliptic coordinates of § 42 the potential at P assumes the various forms

ch v + 1 ch v + 1 sh v p dv

°gehv — ï~~ °^ sh« " °^chü —1— Xv shv(5)

^ , i 1 „ , ,cht) , 1 ch2 v + 1= 2 sh-1 T— = 2 ch-1 T— = 2 th-1 -,— = ch-1 .. 7, etc.

sh v sh v ehv err v — 1

The Stokes function at P of the line AB is

(6) 27T(PJ.-PP)=47racosw,

an expression much simpler.

III. The conical angle of the ring as seen from F on its axis is

(FO_FO\_FO x area

V' \ Fa FA)~ FA-Fa-\(FA + Fa)'

and when the diameters ab, AB are nearly equal, this may be replaced by

,Q FO x area(8) FAs ■



So also for any strip cut out by two radii ; and if the radii make a small

angle dd, the same expression holds when the plane of the strip is tilted about its

mean radius O A through any angle, provided FO is replaced by the perpen-

dicular FM on the plane, and the expression becomes

(FM FM\ ,A(9) {Fa-FA)dd-

IV. For a narrow parallel strip such as QQ", of breadth d.y, and PB the

perpendicular on it from P, the conical angle at P is

PM~t pni? pn-7?w PMxavea/BQ BQ"\(10) ==*(«* PQB - cos PQ B)dy = pj¡..Qc, [p| - p|> )•

47. Considering that V + oil is composed of complete I. and II. E. I.'s, the

complete III. E. I. which arises in the determination of a will serve at the same

time for V; and according to the method of dissection we shall find it depend

on one of the five following forms, which we shall represent in the following

notation, in which the III. E. I. is not restricted to be complete ;

"aA coa 8 + a2 bdd Ca-MYbd8f aA cos 8 + a2 bdd fa-.

MQ2 PQ'

(2) a(PY),f1 PY2PQ'

a A cos 8 + A2 + b2 bd8raAcoad+ A2+b2bdd(3) a{PZ) = j-1+r+-TQt

where PZ is the perpendicular on OQ, and PZ2 = A2 sin2 8 + b2 ;

f a coa 8 (a coa 8 +A) bd8 fQN-MNbdd(5) a(IB)-J- pR2 -pQ- X ~P~lF~PQ'

and these III. E. I.'s are connected in pairs by the ten equations following, as

is verified by differentiation :

m aw- «.(«,- /,- si»- ̂-|| = »s- ™^4

= 1-jT — angle between planes PQY, P QZ ;

a(MQ)-a(PZ0)

r,,„r^ r,,T,„, T . ,MZPQ MNMPa(MQ)- a(PZ) = 72 = sin-1 ̂ =-3^ = cos-1 pz~MQ

= angle between MPQ, QPZ ;



(9)

« , »,•« n, nrrs T • , MZ MP , QZ ■ PQil(^)-il(PJr) = 73 = sin-1 FT7MQ = ^-1FYTM%-

= angle between PQT, PQY;

^/t,^x „,„,„ T . ,QNPM XMN PQ(Q) û( JfÇ) + a(PN) = 74 = sin-1 L^ = cos-1 wrw*

= angle between PC7, QMB;

^ a(MQ) + il(PP) = 75 = sin-1 L^ = C08-i j__„

= angle between MPQ, MQB ;

(ii) n^-a^^g^-c»-.™^

= ¿7T — angle between QPP, QPN;

Ciít>t>\ , n/Dx^N r • -i sin0(a.4cos0 + a2 +Ô2)a(PB) + a(PY) = 77 = sin 1 =- -í

(12) _xPMcoa8PQ- cos- pj? pjr

= |7T — angle between QPB, QPY;

»/»» ~,^„ t- . ,Pi7sinc?PO 1a^lsin2ö-62cos(?(i3) il(PP) + il(PZ) =7g= sm-1 pR pz * . cos-. _ ^ pz

= angle between QPZ, ÇPP;

a(pr)-a{PN) = i,-n»->™f±lpP

(14) _, sin 0(aJ. cos 8 + A2 + b2)= cos PY~PÑ

= angle between QPN, QPY;

„„ , sin 0( andeos 0 + ^42 + 62)il ( PZ) + a( PN) = /„ = sin-1 - PZ FN

(15) _xPMcoa8PQ- cos pz PN

= fir — angle between QPA^, QP£.


506 A. g. greenhill: [October

In a complete III. E. I., when 8 ranges from 0 to 27r, the 7 term disappears

or is 27T, and

* = 2tt - ii = a(MQ) = a(PY) = a(PZ)(16)V =27T-íl(PA7") = 27T-íl(PP).

We shall require in conjunction with ii(PZ) the associated integral

V(^2 + 62)(a + ^4cosÖ) bddMS tl(P7) I'V(A'+ b<)(a +A coa V) bdd(17) il1(PZ)=j - pzF' ~PQ'

and then

(m a (PZ) + a(PZ)- f a + ^(A2 + ¥) hde(18) ax(PZ) + a(PZ)-\ - {A2 + b2)_Acos6lrQ^

(i9s a(PZ\ a(PZ\ f a-S(A* + b*) Md(19) ax(PZ)-a(PZ)=j v{A2 + h2) + Acos-e-p-Q,

two III. E. I.'s, with parameter fxF',f2F', such that

(20) -/,*■-—'<rf + *>, ■<#•-.+ ^, + »,.leading to a simple geometrical construction ; and it is found that 27r — ilt (PZ)

is the conical angle subtended by the circle at Px on OP produced.

So too in conjunction with a (PB) we require

(22, iMPSHiMP*)./^-!.^.

(23) ^m-H^.f^fff^^,and ax(PB) is the conical angle at P2 where PXP2 passes through B, or

P2P' is parallel to .4P.

48. Thus with sector elements \a2dd for the determination of V, the poten-

tial dF"at P of the element about OQ is given by

dV_ ra rdr _ Çr + A cos 8 r A coa 8d8

dS-X PQ'] PQ~ X ^Q~

= PQ-PO-2A cos 0th"1 jPqIpq ,

of which the first part is by Lemma I the potential of the strip O Qq'o, between

OQ, oq radiating from Z, and the second is by Lemma II the potential of a

shaving of uniform breadth dd ■ A cos d and length O Q.



As required in integration by parts, the second part of (1) can be written

n d ( Â . n, , a \ a2A2ain2d 12 —n[ A sin 0th-1 ^^ . T>Q ) -

(3)

dd\ PQ + POJ PQ-PO + aAcoad+A2+b2 PQ

od ( a • a 1.-1 a \ FQ PO-aAeoad-A2-b2A2sin2d

_2d0V PQ + POj~ PZ2 ~^Q~

= 2Âain8th^pQa+p0)-P0{l-p¥2-2)

a A cos d + A2 + b2 bda(PZ)+ PQ dd

So also, if dil is the conical angle subtended by the sector element about

OQ, the application of Lemmas III and IV shows that

da_PM PM PM/QZ OZ\~dd~P~0~PQ~ p&ypQ -po)os

(3)pô _da(PZ)_ po dix da(PY)

~ p& dd Pz2 + dd ~ ' ddso that

... dV 7 dil a2 + aA cos d _ d f . . .,, . a \W W+hW-PQ-2Td{Asmdth PQ + PO)

and thence the previous expressions in (13), § 43.

49. The dissection which leads directly to Minchin's integral in the Philo-

sophical Magazine, February, 1894, is made by lines radiating from M.

Denoting the angle AMQ by n,

(1) \MQf dr) = sector element about MQ = i MY- add = \ ( a2 + a A cos 8 ) d&,

and by Lemma III,

,~ I\ FM\ ,(2) da = î-p^jdv,

da dv a2 + aA cos 8 6 dv da(MQ)

(> ~M = d8 MQ2 ~ FQ = d8 d8~""'

so that for any incomplete arc A Q

(4) a = v-a(MQ).

Now with the substitution of § 26 and in addition

(5) MQf=m2(r-t),so that

(6) b2 = m2(tx-r), (a~A)2 = m2(T-t2), (a + A)2 - m2(r - «,),

Trans. Am. Math. Soe. 34License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


then when M is inside the circle

(7) a — A = mV(r — t2), a = \m [(t — tf) + y (t — tf)~\ ,

(8) aA cos 8 = \m2( t2 + t3 — 2t ),

(9) aA cos 8 + a2 = %m2[r — t + V (t — t2-r — t3)~\ .

Integrating round the circle, the angle v makes a complete revolution when

M is inside, and

= 2tt -2Pc' sn 2/T" - 2B(2fF' )

= 2-rr(1 -/) - 2Fzn 2fF' - 2TV sn 2/T7';

hence Minchin's result obtained directly.

If M is outside the circle, the angle v oscillates ; but we must take

(11) a=lmlV(T-t3)-V(-r-t2)],and now

(12) O = 2 j [ V(T-tyT-<g) - 1J 7^ =27r/+2Pzn 2fF'-2Fc anfF',

equivalent to a change of / into 1 —/, as in going from P to P" so that (10)

may be taken to hold for all cases.

As P travels round the circle CP C the quantities c, A, F, E, K, H do not

vary, and / may be taken as the independent variable.

Starting from C in the plane of the disc, where / = 0,

E+c'F V(13) îî = 2tt, V=4a-~—^- = 4aH, -£^= 2(E + c'F).

At C outside the disc in its plane,/= 1,

(14) Í1 = 0, V=4aF^F=4aIf^fIt, ^, = 2(E-c'F).

As P travels to P", /increases from/to/" = 1 — /, so that

(15) il" = 2tt/+ 2Pzn 2/T" - 2Pe' sn 2/T7',

(16) il + il"=27r-4Pc' sn 2/P' = 27T-4Psin f, il-il"=27r-4P(2/P'),

and il" is the apparent area of the coaxial circle through P seen from any point

on the circle AB.



(17) ~p+ A^pr,= 4E-4Fsin2f'-(a + il")sin ^'= 4P- 2flr«nf,

which is Minchin's equation (23), p. 212, Philosophical Magazine,

February, 1894.

When P reaches P,/ has increased to/' =1 +/,

(18) il' = - 2tt/ + 2Pzn 2/P' + 2Psin f,

(19) il - il' = 2tt,

but

(20) V'-V" = 0.

At P'", / has reached /'" = 2 -/;

(21) il - il'" = 4tt, V- V" = 0,

so that V has no cyclic constant as P describes a curve linked with the circular

edge ; a cylie constant like the 47r for il would imply a non-conservative poten-

tial and the possibility of the creation of energy, called otherwise Perpetual

Motion.

50. Dissected by ÇA^nto strips ydx = a sin2 8d8, then by Lemma IV, § 46,

bdx a sinö a2 sin2 8 bd8 da _dil(PAr)(1) da = -p== -pçj- = -pjp- pQ. -dj = ¿8 '

so that x °f § 27 is the appropriate coordinate angle, leading to E. F.'s of AZT

and fK', andn. a2 sin2 8 bd8

PQf*t* a

Jo \ 2a )-9^-) GQ2 - a2 ain2 8

(2)

/

4 sin2 x ° dx

ri+r*tysin2 x

which is reduced to the form A of § 5 by putting

Í8) sin2 y - ^-^ ( ^TLT? V _ !lTÍ»(3) mX- , ^ 2ß J - ,

and this makes

(4) il = 4jr"^-o^)-C^=4^(/ZT) = 27r(l-/)-4Azn/A'.



With the variable y oí § 29

_Z\V_2 _p_a2 sin2 8(5) l-y-pç2i 1-y- PN* '

and denoting the value of Y for y = 1 by F"',

(6) Y' = 4(yx-l)(y2-l)=-4~==-4^2yxy2,

so that from (11) § 29

0 P 2y bVfy^dy ff 1 \2V(-Y')dyX ï=-ya- VT -jyr^y-1)- ~VT

(7)

= 4P(/'A')-2ZT^~^,

in which

(s, w_^_i_(já.). ^-(^.y,from (8) § 45,

(9) dn2/'A'dn2/A' = A2, /+/'=1,

(10) il = 27r(l-/) + 4ZTzn(l-/)ZT—4Arx + r2

= 2tr(l-f)-4KznfK,

as before in (4).

So also by Lemma II

,„„, ,TT . , PQ + a ain 8 .„,„., a sin #(11) dF=d^logp^^0 = asinödöth-1-pc

and as before in § 48 we may transform this into

/19, dF dT a Jx , ,asin<n a2 + aJ4cosö , dil (PA7")

so that, as before in (4) § 48, over the circle,

r2* ad8 dV dV(13) F+Ml-jT (. + ¿«^-.£+¿55.

To connect with the E. F.'s of APand 2/T7', by means of a(MQ),

da da(PN) dIK da(MQ)(14) d8 ~ dd ~ dd dd

(15) iî = 27r-iî(J7Ç),

as in (4) § 49.



51. Dissecting by lines QQ" parallel to AB into strips

(1) QQ"dy = 2a2 cos2 8d8,

of which by Lemma IV

bdy / QB BQ"\ / a2 coa2 8+a A coa 8 a coa2 8—aA cos 0\ bdd(2) m=~FR2[pQ±prj') = y- PQ^ ~+~ ^Fq' )FB2>

we may takeda _ a2 cos2 8 + a A cos 8 6 _da(PB)

(' ~dd = PR2 P~Q ~ dd '

and then integrate round the circle from 0 to 27r.

So also from Lemma II, with the same dissection,

,,x ,Tr ,1 PQ+PQ"+2acosd „, , , 2«cosö _, , PQ-PQ"(4) dV-dy^pô^pç.-1 — ^Myto ■ pl^rpQ=2dyth"1— ^--,

dV df PQ-Pq\ AafA'shtO_ / 1 J \dd dd\y 2A )-4A2- (PQ-PQ")2\PQ + PQ")

fKs i » • 2ûPQ-PQ"-A2+a2-b*( 1 1 \(5) = i«2 sm2 d-pB- [jQ+pq-)

_ 2 /a2+aAco$d+b2 a2—aAcoad+b2\

= (PM-o)y —pfïp—p q— + PQf PÏÏ ) '

and here again, in integrating round the circle we can take

(6) r.f<^-„-^ + ̂ ,and F + oil is the same as before in (13) § 50.

52. Finally there is the integration for Fand il required when the circle is

dissected into concentric rings ; and now d V/dr is the potential of a circle of

radius r, and

(1) dJV=4Fa- = SK-a-,

v ' da r. rx + r2

(2) r-jfu-!*-/^;^*.of which the result is known already by the other dissections, although the inte-

gration appears intractable at first sight.

So also, from (3) § 31

da _ d2V _4ab E _ B2ab 2H - k'2K

( ' la ~ dddb ~ ~r\ f2 = (rx + r2f ff1 '

leading to an integral of appearance still more intractable.



53. Treating the potential U of the solid cylinder in the preceding manner,

as given by a function homogeneous in the second degree of a, A, b,

dU AdU r.dU« 2U=alla+AdA + hl^

the first term is a times the potential of the curved surface, or skin potential

as it may be called of the cybnder, so that

ldZ7 r2n , , b 7„ f* , , It. — T dt

not an elliptic integral ; but dU/dA and dU/db are given in §§ 44, 54 as the

potential of the cylinder due to transverse and longitudinal magnetization.

We shall apply the same method in § 64 to Mr. Coleridge Farr's problem

(Proceedings of the ßoyal Society, November, 1898), to determine the

electro-magnetic force of a cylindrical coil of finite thickness, where a similar in-

tractable integral is encountered, in conjunction with elliptic integrals, which we

consider tractable.

A graphical representation can be given on the Mercator chart of the integral

dW'/ada in (2) by denoting the angle MQP by cb, so that

(3) th"1 -== = th-1 sin cb = log tan ( \-rr + %cp ) ;

then if cb, d are taken as latitude and longitude, and the curve drawn on the

Mercator chart, the area will represent the integral (2).

Or else th"1 (b/PQ) can be expanded in odd powers of b/PQ = b/(rxAco),

and then the integral of each term such as

p dco(4) » = X (Ä^)2^

can be expressed by means of F and P, by means of the recurring relation

(5) (2n-l)c'2Dn-(2n-2)(l+c'2)/Dn_x + (2n-B)Dn_2=0,

starting with D0 = F, Dx = E/c'2.

Mutual induction of a circle and coaxial helix.

54. The III. E. I. arises in the calculation of M between a helix and a

coaxial ring ; this has been carried out by Viriamu Jones in the Philosophical

Magazine, January, 1889 ; Philosophical Transactions (1891) ; Pro-

ceedings of the Royal Society. December, 1897 ; consult also Webster,

Electricity and Magnetism, p. 457 ; Mascart and Joubert, Electricity (trans-

lated by Atkinson) ; and Gray, Absolute Measurements in Electricity, volume

II, p. 313.



It results from the form of the integral that Mia the same as between the

ring and a uniform current sheet in a coaxial cylinder replacing the helix, in

which the linear helical current yh and the superficial current 7 across unit

height of the cylinder are connected with p the pitch of the helix by the relation

(1) 74=27rp7.

For if A7" denotes the number of windings and <S> the total angle of the helix

of height b,

(2) Nyh = by, where ©--, N=~ = -~p.

The potential of this current sheet has been worked out by Minchin in the

Philosophical Magazine, February, 1894, as being the same as the poten-

tial of a cylinder magnetized longitudinally, or of two end plates of the cylinder,

positive and negative ; the expression for the potential of a plate has been cited

already in § 43 above.

We notice now that M is the Stokes function of the two end plates, or of

the cylinder magnetized longitudinally, and that Myh/2trA is the magnetic

potential in the plane xOz at a distance A from the axis of the cylinder mag-

netized transversely parallel to this plane with intensity 7 ; and denoting this

magnetic potential by il'7, and by W the potential of the solid cylinder,

(3) !â = ii'7' 0r Mp = AV;

and.dU r" r2" —aAcosddddz r2" „™, , °

W Aa=-A-dx=Xl -V(a2 + z2) "I -^cos^th-1-^,

the first integral expression given by Viriamu Jones, employing his notation of

a2 and a2 + z2 for MQ2 and P Q2, but using z instead of x, and changing his d

into 7T — d, to agree with the preceding treatment.

The result for Mp is the same whether 2 a denotes the diameter of the helix

and 24. of the ring, or vice-versa as with Viriamu Jones, being symmetrical in

a and A.

Integration by parts enables him to replace (4) by

,x i~i r2"a2A2bsin2ddd r . ^bd8 ,. . ,,.-,.««,(5) ¿n'=Jo a2V{a2 + yy = iJ (-aAcoa8+a2)pQ-\(a2-A2)a(MQ),

integrals discussed already in §§ 33, 49 ; and (5) is the equivalent of the double

integral

r* ft* a2A2 ain2 ddddz r"r2a2dddz

<•> ASl=ll (.*+*, -11 -*T'showing that the magnetic potential il' at P of the cylinder magnetized parallel

to OM is the same as that of the surface magnetized circumferentially with

intensity a sin d.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


It is the second form (5) which is reducible immediately to the standard

elliptic form, and by the substitution in §§ 26, 49, writing

(7) MQ2 = a2 = A2 + 2Aa cos d + a2 = m2(r — t),.

(8) PQ2=a2+b2=m2(tx-t),

(9) b2 = m2(tx-r), (a~A)2 = m2(r-t2), (a + A)2 = m2(r - Q,

and thence from (5)rh dt

(io) Aa'=ib\ ^t2-t)(t-t3)vs%-tt t)v ' 2 J h m2(-T — t) mV(tx — t)

_ ç(t2-t)(t-t3)dt

~ X (r-t)VT '

, C( T — t-T — tf\ dtpM=Aa' = mbJ {t-t3+T-t2-T2;rr-3\-7?

ir—S ■*-** CiV(-T') JL 1\tx-r-tx-t3J r-t VT\-

Hence, on fig. A,

(12)

It, — t b . , It — t a — A

V hd¡=a~rA: -cos *'=A+' * = am 2/F''so that

(13) ^J7= rxb[F-E+ Fc'2 cos2 ̂ - cot ^ cos f B(2fF')~] .

(14) —s = P-P-cotf cos^'^(l-2/)P',

the equivalent of Viriamu Jones's result in (6) p. 198, Proceedings of

the Eoyal Society, 1898.

We shall replace his angle ß by yjrx to avoid confusion with Maxwell's

modular angle ß, and then

.-, „„,,-., tan ifV' 8-ir, , i/(aj4)(15) f, = am 1- 2/) P', sin f = - —V = —r~l - > cos t = ■ / , i n »V/Tl v J ' ri tani/r a + J.7-2 Tl J ( a + J. )

so that the angle \jr'x is constructed on fig. A by drawing the circle on the

diameter AM cutting the axis OF in G, and then drawing OPx perpendicular

to the radius GO' of this circle, then the angle OAPx or O GO' is i/r¡.



55. With the notation of § 47, equation (5) § 54, is written

(1) 2Aa'+ (a2-A2)a(MQ) = f (a2-aA coa d)^.

(2) 2Aa' + (a2-A2)a(BY)= f (a2 - aA cos d)~ - (a2 - A2)I3;

and in a complete revolution of d from 0 to 27r,

f* MQ2 + a2 - A2 bdd r dd f a2 - A2 bdda(MQ)-Ja -^gr- -pQ=bJ p-Q + J -MQt-pQ

7»

= 2P + 2P(2/P') = 2Pc' sn 2fF' + 2B(2fF')*i

equivalent to the result in (10) § 49.

Also 73 vanishes in a complete revolution of d, and from (16) § 47,

il(PF") = «I>=4P(/7T'),so that

X2" bdd(a2 - aA coa d)p^

(4)8a bK _, , . , T_ „,

= ^+-^ + ^(r1 + rs)(K-H),

in which2a 1

(5) -= ~ = dn fK',v ' rx + r2 ch v J

(6) pM=Aa' = b(rx+rf)Kdn2fK' + b(rx + r2)(K- H)

-2(a2-A2)B(fK'),

in the form given in (5) § 44, the quadric transformation of (13) § 54.

This again, by means of (2), (3), § 17, can be transformed into

(7) . M .f, = K-H+Cn^7C(l-2f)K'.K ' (fj-f r2)@ sn 2/AT v "' '

In the indeterminate case of 2f= 1 and C( 1 — 2f)K' = oo, we have A = a;

but returning to the integral in (10) § 54, and putting t = t2,

(8) pM= Aa' = ^ ji^7t^dt= bmV(tx - tf)(F- E) = rxb(F- E),

(9) ^=rx(F-E) = (rx + r2)(K-H+kK),

and this is the coefficient of self induction of the helix on a final turn of itself.



Numerical calculation of the mutual induction.

56. The numerical calculation of M for given dimensions a, A, b now

requires the five operations following, in the numerical tables of F. E., vol. II,

Legendre, and it is useful to check the calculation on fig. A, drawn to scale.

I. Calculate yjr and yjr' from

b , b(1) tan yjr =--j , tan yjr =--. ;K ' r a — A' r a + A'

and then

rx = (a + A) seci//, r2 = (a — A) aecyjr.

II. Calculate yjrx and the modular angle 7 from

cos ^ sin yjr'(2) sin y, =-p, cos y = ~—r •K ' T l cos \jr sin yjr

III. Calculate/by Legendre's Table IX from

IV. Calculate zn(l - 2/)P' by Table IX from

(4) zn(l - 2/)P' = E(ß\ c) - (1 - 2/)P(|7T, c').

V. Look out

(5) P=PQtt, c), P=P(Jtt,c).

When f is one of the rational fractions of (1) § 4, we can express c, c',

sn 2fF', zn 2fF', ■ ■ -, as algebraical functions of a parameter, leaving F and

P as the only transcendents, and these are tabulated by Legendre in his Table

I to an extra degree of accuracy.

In the numerical application of Viriamu Jones, Philosophical Trans-

actions (1891), Proceedings of the Royal Society (1897),

2a=21.02673, 24 = 13.01997, 25 = 5.02480 (inches),

a- A =4.00338, a + A = 17.02335 , 5 = 2.51240;

and working with four figure logarithms we find i/r = 32° 3', yjr' = 8° 24',

7 = 74° 1', -f j = 58s 58'; or to the nearest degree, 7 = 74°, fx = 59°, thus

avoiding proportional parts in Legendre's Table IX ; F = 2.70807,

E= 1.08443, and P'= 1.60198, E(fx, c') = 1.04123, 1 - 2/= 0.6500,

/= 0.1750, tt/= 0.54985, E' = 1.54052, (1 - 2/)E' = 1.00134,Ey¡rx= 1.01847, zn(l - 2/)F' = 0.01713, Pzn(l - 2f)F' = 0.04638,

A(l - 2/)P'= 0.50347, cot -feos f'AÇl - 2/) P'= 0.7952, F—E= 1.6236,M/rx% = 0.8284, @ = 201tt, r, = 17.21, M = 9001 inches, as against 9028inches, given by Viriamu Jones.


1907] THE ELLIFriC INTEGRAL IN ELECTROMAGNETIC THEORY 517

This rough calculation is retained for the purpose of showing the term requir-

ing extra calculated figures, compared with the parts where fewer figures are re-

tained for the same degree of accuracy ; thus of the three chief terms of M/rx ©,

the first F — E = 1.6236 is about double the second -nf cot yjr cos yjr' = 0.8638,

while the remainder cot yjr cos yjr'Fzn(l — 2f)F' = 0.0734.

The modular angle must thus be determined with extra accuracy for the

determination of F— E; and in this case raising the modular angle by one

minute to 74° V will make F— E= 1.6258, and raise M to 9024.

A diminution of 0.1 per cent, in f in the second term will raise M about

0.05 per cent, to about 9028.5, without affecting the last term appreciably.

Variation of the induction due to change in the dimensions.

57. Viriamu Jones calculates the effect of small error of measurement

dA, da, db in A, a, b in giving a change dM in M, such that

.__ f .ôM\dA / dM\da (BM\db(1) ^=(âz)ir+(aäa)¥ + (6^JT'

so that 1 per cent, change in A, a, b gives a change in M of

A dM a dM o dM

( ' îôô bJ ' loo âd ' ïoô lb;

and their sum is M/100, because M is a homogeneous function of the variables

A, a, 6, and of the first degree, the total helical angle © or number of wind-

ings © -t- 27T being kept constant. But if the pitch p of the helix is kept con-

stant, M is of the second degree in A, a, 6.

Differentiating the double integral expression for M in (4) § 54,

„, © rb C2lT Aa cos ddddz

(8> M-J,l - *<.* + *) ■and supposing the limits of d to be 0 and 2tt in the sequel, we have

dM © rr — acoa ddddz © f r(A + a cos d)Aa cos ddddz

(4) dA = bJ J ~V(a2 + z) +b) J ~(a2 + z2y

rN , dM „ _ f (A2+Aa cos d)Aa cos ddd ^ t. fa2+aA cos8 dd(5) A M =M+®J \--2v/(g2;+62) - = ®A2J - -jjfqr-pQ,

and similarly

dM ., „ f(Aacoad+ a2)Aacos8dd „ , faA cos 8 + A2 dd(6) a^=M+®^-a2V(a2 + b2)-= &a J ' -JZÇ^^PÇ'



so that

dM . dM .__ r-aA cos ddd(?) alïa+AdA=2M-@J -PQ-'

while, keeping © constant,

/ox i tir r, C C—aA cos 6dddz

(8> lM=®xx-7w+-*r>TdM ._ _ r-aA coa ddd

(9) 6^+J7=©J —Pc—,and then

dM dM , dM ,

a verification of the homogeneity. Also

1 dM 1 dM Ä f dd , © M „ simlr'

1 dJ7 ldJf ra2-A2 dd 4B(2fF')

' ) AdÄ~ä~dl~ J MQ2 PQ- ^ '

(H, ¿«.î^-vr-^^H-iail^,

and we see now that the conditions are satisfied that this JZof Viriamu Jones

is the Stokes function of Minchin's pair of end circular plates of density

7 = =b 1/2-rrp, corresponding to a helical current <yh = 1, and

dM tdM TdM „(15) 2M=a-^ + AM+b-dJ + M,

(16) ^J7= -2a2-A(l -f)K + 2A2-B(fK) +b(rx + r2)(K- H),

in agreement with (6) § 55 above.

58. Thus the magnetic components for the helical current yh, axial and

transversal, are, for magnetic potential il,

W ï —A M= -4^(/^')= -7(27r/+47rzn/7T'),

and, now keeping ^> constant in (8) § 57,

di! yh dM 7 p — a^l cos 8dd.1 rdA~2trA db~ AX PQ

(2) /. + *W m A..z„wK-H ,.. K~H



With 6 = oo we must take tx = go, and the integrals in (10) § 54 reduce to

>t2—tt — t3 dt

(3, ix-ja-*^ T_, Ai.h_t.t_k)

= m2íj[t-t3+,-t2^-r^)v{^_t_t_t¡)

(4)

= m2^-^3 + r - t2- V (r - t2r - t3)^7r

with

(5) 4aA = m2(t2-t.f), (a ~ A)2 = m2(r - t2), (a + A)2 = m2(r - t3),

so that

(6) pM=irA2(A <a) or 7ra2(^4>a).

Otherwise, with 6 = oo in (5) § 54,

„ ,~, C2" a2A2ain28d8(7) pM = Aa = I -i-ñ—-.-ä—,—72 = T-42 or ira2,w r Jo « + 2aJ. cos 8 + A2 '

employing the substitution of § 26.

Thus for the infinite helix, in the plane of the end .4P,

da 7,

P(8) _JL = _ _rA=_27r7 or O,

and at M or M',

m S-^t5 « *»«-■*>.i,im i 0M 0A(10) Ä.--Q2 or -uw.

Thence by subtraction, for a helix extending from b to infinity,

(11) ^-4tmí(/K"),

dû t r T7-77 K-Hl<12> d^ = 44dMT^7)^--V"J'

with 2/ > 1, according asi>a.



Potential energy of two coaxial helices.

59. There remains still the evaluation of the potential energy of two coaxial

helical currents or of their equivalent cylindrical current sheets, given by V.

Jones in §13, p. 202, Proceedings of the Poyal Society, 1897, and

this is the same as the mutual potential energy of the two pairs of equivalent

end plates. Viriamu Jones shows that this energy depends on the integral

(1) F(z)=A^£\oa8ddfi(z),

(2) f(z)=zlog[^l+Ç) + l]-V(a2+z2), f'(z)= [loëyj(l+^)+l],

(B) F(z) = Mz+ r~AaC08eddV(a2 + z2),Jo P

and the second term in F(z),

m3 Cs

= jf((t2-t-t-t3)-VT^T)

dt'V~T

(4)dt

V7"

the same integral as for G in § 32.

But it is the force between the two helices which alone is required, and this

is given by dF(z)/dz, or by the change in F(z) due to a small relative axial

displacement of the helices, and as this is equivalent to the removal of a circu-

lar element from one end to the other of a helix, this force depends only on the

difference of the values of M between one helix and the two circular ends of the

other helix ; and this is calculated by the preceding analysis, which gives the

value of M as a function, say M( 6 ), of b the height of one helix say of radius

A, when the ring of radius a is in the plane of one base of the helix.

When the circle is at a distance z from the mean section of a helix of height A

(5) M=M(z+\h)-M(z-%h);

and since M(— z) = — M(z), this makes M= 2M( |A), in the mean section.

The hydrodynamical analogy is complete, M being the Stokes function, and

V(z + %h) — V(z — JA) the velocity function, V(z) denoting the potential of

a circular plate, due to a uniform distribution of source or sink over the plate,

thereby producing a streaming motion circulating through the helix, a line of

flow having M= constant : thereby we gain a physical conception of the lines

of magnetic force inside and outside a solenoid.

Maxwell's figure XVIII can be utilised by the method of superposition for

drawing these lines when the ring is made to expand into a cylinder by pro-

gressive axial displacement.



Exploration of the electromagnetic field.

60. The exploration of the electromagnetic field of the ring current is given

by Maxwell, E. and M. § 702, and in fig. XVIII by drawing the twelve sur-

faces il = 27rf, taking

(1) /=(0,1,2,3,...,11,12)155;

equivalent to direction at the ring on a clock face at P, for every hour and half

hour from XII to VI, and again from VI to XII.

Any surface il = 27r/' will cross the axis Oz at a point H such that

(2) sin yjr = sin OBH= |g= 1 - g = 1 -/,

so that H is determined geometrically by drawing a circle center A and radius

(1 —f)AB, cutting the circle on diameter AB in Z7, and joining BH cut-

ting Oz in H.

Thence at O, where f= 1,

dil_27r(l -/)_2tt

~db~ a(l-f) _~o~'

the galvanometer constant for the ring.

Practically we need only draw the surfaces from f = 0 to \, or between XII

and III o'clock, as the remainder from III to VI can be inferred by the theorems

in (25) that OM- OM' = OB2 for points P, P on a line through P.

In another method of exploration we can utilize the analytical results given

in Philosophical Transactions, 1904, and determine the value of il, V,

and Min §§ 34, 42, 43, 44, 54, when 2f is one of the simplest rational fractions,

such as

(3) 2f-l 2 1 1 l^3 1 1W *J— J-> 3> 2' 3' 4 ' 5' 6' "'■•

In this method it is simple to work with the elliptic functions to modulus c

and argument 2fF' and to use the quadric transformation to obtain the result

to argument fK, if required.

A curve along which 2f is a rational fraction is an algebraical curve, passing

through A at an angle /7r with Ox, and having Oz as an asymptote.

We proceed to trace a few of the simplest, and for this purpose it is useful

to employ a stereographic chart with poles at A and B, and center of projec-

tion the antipodes of O, in which case, as in § 25,

(4) A = tani\ = tha, c = tan (45° — \X) = e~2a,

X denoting the latitude, the angle FOL on fig. A.



Exploring the values of il, V, M at the principal divisions of the clock face :

1 — CXII o'clock, /"=0, 6 = 0, yjr = 0, x = ak = a z.-,,

1 + c

í1 = 2tt, V=16aF+cF, M=0.1 + c

VI o'clock, y= 1, 6 = 0, yjr = tt, x = y = a ^->,

il = 0, F=16ajS'~Cf, J!f=0.J- ^~ c

/

Illo'clock, 2/=l, -^r=i7r, x=a, z=2a —, sini/r'=c', i/»,= 2am|P'—^7r,

«r,» Tr , H (2H \ M - P-P P-Pil=7T-2Pc', F=4a-7rz = z(—— tt), s = 2a-= z-—,

c \ c ) © c c

on reference to (9) § 55.

II o'clock, 2/" = f ; and (see Philosophical Transactions, 1904, page

261) in region B, 3 >p > 1,

(y-l)3(z» + 3) 2 (p + i)3(-p + B)

Wp ° - 16p

3 — v B — pzn |P' = —ç£ Vp, zn §7" = -~^ ,/p,

» — 1 p — 1cos yjr = en §F' =-j, cos yjr' = dn §T" = —~—, sec-v|r — seci/r'= 1,

along the curve, analogous to

yjr + yjr', or tan yjr + tan -^»-', or cos yjr + cos yjr', • • • a constant,

a circle, or parabola, or magnetic line of force, • • • in biangular coordinates yjr, yjr .

2I o'clock: 2/=J, sin¥=sn JP' =——=,

1.30, 2/=i, | = amii", 8^^ = ^^^, sin yjr'= V(l -c),

tan2*-ten2*'=l, ^-^2- (-^? = 1,

zn ÍJ"-Í(1-C), Íl = f7r-P(l-C)-2P1/(1-C).



XII. 45, or II. 15,2f= \ or £ (Philosophical Transactions, page478),

i>27>v/2-i, ^c = iy--Py

cos2^'=dn2iP' = (i±Äi_-_Z), ̂ ^Jl±ÉlíZ^±¿lt

(* +7)(1 —p) 1°>

and a change of p into — 1/p will change 2/from ^ to f.

XII. 36, I. 12, I. 48, II. 24; 2f=\, f, f, *, and the results in Philo-sophical Transactions, page 264, to be consulted.

XII. 30, II. 30, 2/= \,\; consult page 283.

61. In an apparatus constructed by Professor Ayrton and Viriamu Jones,

2,4 = 8, 2a = 13 inches;

and then for the height b we shall find in correspondence with 2f,

2/

b

0 1 1 1 2 3 1"' Ï' 3' 2' 3"' 4' x

0, 1.36, 1.47, 2.57, 4.59, 9.79, oo.For if

(1) V-i, i = 1^-2.574,

(2) 2/-,, (i=£ )'_ «-«,(!_ V)r- í£±i!£fctil,

jj = 2.8322, -A-, = tn \F' = —& , & = 4.593,7 a — A ó p — 1 '

(3) 2/W, l^.dnfP'-^, ,, = 2.9428,

6 2= tn IF' = .. , , 0, , b = 1.471,

a A 3 "i/(^-l-j» + 8)

(4) 2f= | dnUP'=^^=4L6_(I±Zm_-_iL)W ¿7 f> an 4^ (a + .4)2 441- 4^

a quartic for p having a root p = 0.43 and making

<s~*l(-—p\ .0.719, c'= 0.695,

cos2 yjr' = dn2 £P' =|11C2= 0.547 (Vp/ = 43o)f

0= (a + A) tan ¥' =9.79.

Trans. A •<. Muth. Hoc. 35License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

524 A. G. GREENH1LL:

Similarly for 2/= £, o = 1.364.

[October

62. To determine the proportions of the apparatus when the modular angle 7

is raised to 75°, and we take 2f= \, as in fig. A, we put

(1)(7-l)3(P + 3)

= sin2 75° =2 + 1/3

l<op --" '" 4

of which one root is p = — |/3, but this root leads to

(2) sn£P=1/3-l, dnJP=J1/2.

But with the other real root,

(n + i)2 /^2 + iy 9Q,8Rfi1

*-73—=lf2^ïj = 2-948661'

csc yjr = sn \F' = } (/» + 1 ) = 1.97433

(3)

(4)

(5)

(tf> = 30°26'))

sin yjr' = c' sin yjr = ~l P + 1.-P + 34p

= 0.1319 (V' = 7°34'),

(6)

CO

sin i/tj = sn §P' =2 1/7

( Vi = 60° 26'),

(<=13°13'),

JO + 1

cos i/r; = dn IP' = I (j> - 1 ) = 0.97433

(8) rx = (a + A) secf ' = (a + A)dn }T" = (a + A)^p _ 2]Pp + g,

(9) zn(l-2/)P'=zn|P'=^(3i,i-^),

(10) 7^0= ^-^-4^>-U —6/'

(11) P= 2.768063145, P= 1.076405113, P-P= 1.691658032.

These angles are contrasted with those of the apparatus of Viriamu Jones in

the table following.

2/

sin- c = 7

sin-1 k = ß

^1

A ¡a

J = 0.33

75°36° 4'

30° 26'

7° 34'60° 26'13° 3'0.63196

0.35

74° 1'34°36'32° 3'8° 24'

58°58'13°36'0.61921



63. If the windings of the helix are not complete, or. if the current is consid-

ered through a fraction of the circular ring, the theta functions are required or

the pseudo-elliptic algebraical III. E. I. can be employed for a rational value of

the fraction f the simplest cases of which will be found worked out in the

Philosophical Transactions, 1904.

Lord Kayleigh has shown, in a Report to the British Association

(1899), that all error in extreme cases of this assumption as between two coaxial

helices is eliminated by taking an average for different symmetrical positions.

One object of the present memoir is to see if by a slight change in one of the

dimensions of a helix, say the height o, designed for the experimental determi-

nation of the ampere, it would be possible to make 2f one of the rational frac-

tions given above, and so simplify and check the numerical work.

Or we may construct the apparatus so as to make the angle 7 in fig. A or

ß in fig. B an integral number of degrees, so that the value of P and P or

K and H can be taken out of Legendre's Table I and IX without use of

proportional parts, and utilise all the decimal figures.

Potential and Stokes function of a solid coil.

64. To solve Coleridge Farr's problem for a solid cylindrical coil, we have to

determine the potential W and Stokes function P of a series of coaxial helical

currents or equivalent cylindrical current sheets filling up a solid cylinder, inte-

grating with respect to the radius the expression for a cylindrical sheet.

It is simpler to determine W for a coil stretching in fig. A from the plane

.4P to infinity on the right, as this is equivalent to integrating over a single

circular plate .4P of superficial density 7 = yh/2-irp (qualified as shown in (8)

by the factor a — r, r referring to any internal radius OQ'), the influence of

the other end plate at infinity being insensible.

In § 48 we have found for the potential F of the plate .4P of unit density

(1) ^¡ = PQ - PO-A coa 8-2th->pQa+po,

the equivalent of cPW/dadd, and integrating by parts with respect to a,

(2) ^= £ (PQ - PO) da -A cos 8J 2th-1 pg^pQ da

= (la + §4 cos 8)PQ- (a + %A cos 8)PO

- 2th-1 pQ ° pQ(M2 - iV + aA coa 8 + \A2 cos 2d).



Integrating again with respect to d from 0 to 27r, the term

(3) t-f^THSTFÖ»

is not an elliptic integral, and so considered intractable, like the integral (2)

in § 53, to be denoted by Ix ; but

j"aA cos d ■ 2th-1 pQ^-p-ödd=a f(l- p^(aA cos d+A2+b2)~

(4) -a1/(J2 + o2)J(l-¿)do

= C(aA cos d+ A2 + b2) a -^ + ao [( 2tt - il (PZ)] - 2-n-a y (A2 + b2 ).

/„.» „ , , « in .o f sin20d#A2 cos 2d - 2th-> j^^rpo d6=- aA J -PQ7

(Ö)+ (¿2 + O2)J(a + ^cos0)~-61/(^2 + &2)íí1(PZ),

so that IF is composed of

(6) (W-\A2)I, -a6[27T-il(PZ)], lbV(A2 + b)ax(PZ),

and I. and II. E. I.'s, amounting to employing the theorems of § 33,

2IF=(62-^2)7-2a6[27r-il(PZ)] + U^/(A2+b2)ax(PZ)

(7)+ — (a2 - A2 - lBb2) + BarxE.

The potential IF may also be considered due to the circular plate .4P, sup-

posing the density at any interior radius OQ' = r is a — r; and then, expressed

as a double integral

d W r C rdr dd(9) ~j— = I j -j57y- — potential of the end plate of uniform density

/dd (a2+ aA cos 8)p-Q - b [2-rr - ii(PZ)] ,

dW_ f r —(a —r)(r coa 8 + A)rdrd8(lü) dA=J J ' "ïY"



dW_ f r-(a-r)rdrbd8

(11) ~db -J X ' ~l?tf ;and

dW AdW J.dW OTT^

(12) alte+A-aTA+hllb=2W<

the homogeneity condition of IF, of two dimensions in length.

65. Other double integrals required in the calculation are

(i) 2,-a,^).//^,

the apparent area or normal attraction of the disc AB ;

(2) ni(pZ)_r/!4iw^?,

C fdrdS(3) '-JJ PCF-

the intractable integral (3) § 64 which is the potential of the disc .4P, with

density r~l ;

r fr cos ddrdd Ç Ç (r + A cos 8) cos 8 — A cos2 8

J J PQ'3 =JJ PQ'3

/•/ 1 1 \ rAeos28/a + Acoa8 Aeoa8\w =}\pö-pQrsede-j -pzt-y—pQ—po-)de

a r d8 V(A2 + b2) a(p7., /tA%. m feos 8d8= Aj PQ-A^--ílÁPZ) + v(A +b)\ -p^2-,

from (17) § 47, and the third integral vanishes ;

AdK„2W-a---b--._. d.4 " du db( ' aF

= - $A2I+ Îbl/(A2+ b2)ax(PZ) - — (a2-A2+Bb2) +arxE,"i

r*S dP 9 4dW dP 0 AdW(8) _r¿r=_27r^z, ^.2^-gg-.



66. The Stokes function P can be found by an integration of Viriamu

Jones's M, but an infinite constant comes in when the coil stretches from b to

oo; so returning to a coil extending from 0 to o, or from AB to the parallel

plane through P,

d2P dpM a2A2 sin2 8 b

(' dädd~~dd~~ MO2 PQ'

and since

(2) a2 = MQf - 2A cos d(a + A cos d) + A2 cos 2(9,

' a» • 2û Chda a as ta aÇa + Acoadbda= A2 sin2 d I ^7~ — 2 A2 sin2 d cos d I -¡r^,-=—=

J BQ J MQ2 PQ

+ A2ain2dcoa2dfw^-bda

+ A' sin2 d cos "¿d \ lir7^

(3)

=A2b(1-cos 28)th-1 pQapQ + 2A3 sin28 cos 8(thr1 ~ -th.-1 ^

— A3 sin 8 cos 2(9 \lt - tan"1 ( —,— tan d\\,

where I2 is given in (7) § 47, and the integration of this term by parts with

respect to 8 will depend on a(PZ) and ax(PZ) ; and P is thus composed of

\A2 hi and I., II., and III. E. I.'s, which can be evaluated in the preceding

manner.

The intractable integral

C~ , , a -,* C . ,o+ oleoso ,„ C , , .4cos0 ,„(4) i=j 2th-ipô7TP-ôd8= J th--^-de- J th-wd*,

of which the second integral vanishes.

Putting (a + A cos 8)/PQ = sin cb', so that ep' is the angle between PO and

PZ, and drawing the curve on the Mercator chart connecting latitude cb' and

longitude 8, the area will represent 7, as before in § 53 for Ix.

For a flat coil in the plane ^4P,

— 2-rrAr cos 8 drdd

<6> F-fJ PQ(6) ~--2irAo>e(PQ- P0) + ,A'(i+ca¡<ie)t\-> pffpQ,

so that

(7) P = £7T^27and I., II., and III. E. I.'s.


the elliptic integral in electromagnetic theory 529

Legendre s discussion of the oblique cone.

67. The method of discussion by Legendre of the surface of the oblique

cone in his Fonctions elliptiques I, p. 329, may be resumed here, because his

ep, the angle of the sector of the developed surface, and il the conical angle of

opening of the reciprocal cone are connected by the relation

(1) <f> + i! = 27T,

so that one calculation implies the other.

Fig. D.

Adopting Legendre's notation for his oblique cone on a circular base, figures

C and D, and considering it as the reciprocal of the cone discussed previously

in figs. A and B, then PM the perpendicular from the vertex P on the plane

of the circular base AB making an angle X with PC the bisector of the angle

APB, and the edges PA', PB' of the reciprocal cone making an angle X' with

PC, and putting (Legendre)

(2) AO= OB = r, OM=f, MP=h, PA = a, PB = d,

sin P.4 C= sin ( X' + X ) = -,(3)

(4)

(5)

(6)

CO

(8)

in Legendre's notation.

sin PB C

cos 2X =

cos 2X' =

cos X sin X' =

sin ( X' — X ) = -,v ' a

r2 -f2 + A2

cos ( X' + X ) =

cos(X' — X) =

2/A

r-fa '

r+f

aa

:_f2

2X =aa

A2

aa

h(d +a)

2ad '

tan X d — ai-^-< = -r-— = cos d,tan X a + a

. <, , 2rAsin 2X' = —j ,

aa

h(d — a )sin X cos X = —^-s—;—',

2aa


530 A. G. greenhill : [October

If Pq is the generating line of the reciprocal cone perpendicular to the plane

PQY, making an angle ß with PM, atid c denotes cos AOQ,

n MY r — fc sin 2X' - c am 2X(9) cot ß = ±>H7 =-ÎT---¿vv-ÎT-,v ' PM A cos 2\ — cos X

cos ß cos 2X' + sin ß sin 2X' = cos (2X' — ß)

(10)= cos ß cos 2X + c sin ß sin 2X' = cos ß',

if ß' denotes the angle between PQ' and PM' the perpendicular on the other

circular section of the cone in fig. D ; so that

(11) ß + ß' = 2X',

or the sum of the angles which PQ' makes with PM, PM' is constant, and

these are the focal lines of the reciprocal cone in fig. A ; hence the fundamental

property of a sphero-conic.

68. Legendre denotes the angle of the sector APQ when developed by ,

and the angle A O Q by «, so that

(1) PQ2d to our form, put

2

2

PQ2 , r2-2rfc + f2 + h(4) m2(er-s) = p-^ = sec2 * - {rZfi/+K

(5) m2(s2 - s) = JÇ = tan2 f = ^^"^

(6) m2(a-S2) = l.

Then, as before in § 29,ad . /(a' + a) — (a— a)c\2

(7) m2K - .) = ^cos2 ^~^4-¿-LJ ,

aa' . „ /(a' + a)c — (d — a)\2(8) m2(s-s3) = 1? sin2 X { ^_^L_^--' j .

Putting s = s2, c = ± 1,

aam2 ( sx — s2 ) = ,Y cos2 X, ?n2 ( s2 — s3 ) = -p- sin2 X,

<•) aa» y2

m2(Sl-h) = -tfl mÍ(S!-S2)(S2-S3)= p»



(10) sin2X = S?-*3=A2,Sl ~" S3

so that X is the modular angle,

(11) m2(sx — a) = -p-cos2 X — 1 = -p-cos2 X',

(12) m2(a — s3) = -p-sin2 X + 1 = -psin2 X,

(13) JmV(-2) = ~.

Again from (11) § 67,

tan (ß — X') = tan (X'- ß') = tan i(ß-ß')

(14 ) c cos X sin X'— sin X cos X'= tan X ;=tanx^;+a)v(a'-«> /^,

a + a—(a — a)c \sx—seos X sin X — c sin X eos X

so that as before in § 35,

,in j0 fhdc ¡s — s ds

(15) dP = {r-fc)2 + h2=dUri y¡sf--¡-7(^sl-s-s-s3y

rdco — rdc —frde r ds

(lb) p'r= PYV(l-c2) = PY2-mV(s-s2j=~hmVS'

v ' J„3 cr — s A i/»S J tr — S V«

We notice now that in Legendre's notation, with

(18) cos 6 = , —, and tan \co = tan |# tan %cb,

equivalent geometrically to a change from the excentric anomaly co to the true

anomaly 

' a + a — (a — a)c 1 — eos co eos ti T

/rmx Is — S3 -. ± ls~ S3 C0S(P(20) A-? = tan X eos cb, ../-*- = —-ffy ' \sx — s Tt \s2 — s3 Aeb

and the elliptic functions here are a quarter-period out of phase with the pre-

ceding in § 26, and

(21) <f>=am(l-A)7T.Also

(-22) °n'-^g' . g~'» Sî< = *1 _?__ . 4ga> . sin2 e^ ' dn2/A' cr — s3sx — s2 ad sin2 X' cos2 X (d + a)2 '

so that

(23) 6»=am(l-/)P'.


532 A. G. GREENHILL : [October

Stereographic projection of confocal sphero-conics.

69. Describe the unit sphere with center P, cutting the focal lines PM,

PM' in A, B and PQ' in C, so that in the spherical triangle ABC (fig. P)

(1) ^4P = c = 2X, BC+ CA = a + b = ß+ ß' = 2X',

and the locus of C is a sphero-conic for

constant a + b or X' or f, the orthogonal

confocal sphero-conics having a — b or A

constant.

Projecting these confocal sphero-conics

stereographically on the tangent plane at 77,

the middle point of A'B, where the spheri-

cal triangle ABC is colunar with ABC,

and putting CF=ti, CFB = yfr,y/e can

put, according to Burnside, Messenger

of Mathematics, vol. 20 (1891), p. 60,

tanJ6Vi = y/c'sn(?; + ui, c') [c' = tan Hff —«)]>

and obtain the orthogonal system of curves given by Holzmüller in Isogonale

Verwandtschaften.

Here FA = 90 + Jc, PP = 90 - \c,

cos a = cos d sin \c + sin d cos \c cos cb cos o

= — cos d sin Jc + sin d cos Jc cos ep.

sin !( 6 — a ) sin J ( 6 + a )

(2)

(3)

(4) 2 cos d sin Jc = cos a — cos b,

But, writing w, w for w ± ui

(5) tan J# = c' sn to sn w ,

cos 0 =

(9 =

sin \a

1 — c anwanw

(6) cos d =

1 + c'anwan w'

1 — c sn2 v 1 + c sn2 w£

1 + c sn2 « 1 — c' sn2 ui '

so that replacing tj and m by fF' and AP, and employing the quadric trans-

formation,

dn/ZT'(7)

(8)

(9)

(10)

(11)

(12)

cos d =

sinJ(ó + a)=dn/AT',

sin 1(6 —a) 1

dn2AAÏ'

cos \ ( b + a) = 7T' sn/A",

cn2A7T M.r= di2Ä7T=sn(1-2A)^sin \c dn 2AT¿

sin |(6 - a) = AT sn (1 - 2A) K,

cos 1(6 - a) = dn(l - 2A)P,

A' = cos \c.A = sin \c,License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Thus A = J makes a = b, and the stereographic projection of the great circle

GH is a circle, the coalescence of two branches of a quartic curve in Holz-

müller's diagram.

From Napier's formula

/1Q. cosi(^-P) tanJ(a + 6) dh/A' 1(Id)

cos l(A + B)~ ten Je «sn/ZT' ~ cn(l -f)K'

we deduce

(14) ten U tan |P -{'2^-9^''and from

sin|(^-P) _ tan|(«-o) _ ¿_'jn_( 2A-1)TT(lö) sin^ + P)- tanjc " dn(2A - 1)K ~ ~ CD Z/Ui

we deduce

, r, 1 - cn 2AP(16) tan \A - tan \B = 1+<a|24jr

,, . l-cn(l-/) l-cn2A(17) tan21.4 = T--W—44 ■ «—,-sí »v ' 2 l + cn(l—/) l+cn2A'

n«\ m A cn(l-/) + c°2A sn(l-/)sn2A(18) cn^ = r+cn(1+/)c-^, s,n^ = i+Cn(l_/)cn2A'

(19) lrf|jt.i=5iJrZi.J±-«iv ' ¿ l + cn(l— f) 1—cn2A

cn(l-/)-cn2A sn(l-/)sn2A(20) cosP = 1 v ,/; „—=,, sinP = ^-V „ J '.-m,v ' 1 — cn(l — /)cn 2A 1 — cn(l —/)cn 2Aand so on.

70. If N, N are the poles of the circular sections of the cone, or the foci of

the sphero-conics of the reciprocal cone,

« Sin m~ tanT(a\b) — • — (1 -/)**.

and we find by Spherical Trigonometry

(2) cos CN, CN' = cos CE cos EN± sin CE sin PA sin CP.4

and^v,-, cos i(a + 6) cos i(a — 6) „„ .„ „. %

(3) cos CE =-*v ' . 2V-y = sn/dn(l-2A),COS g-C

•m.,- i/sinssin (s —c)W C0S ̂ - £»*»(«■+*) - -<!-/).

(5) sin (7Psin CP¿ = v/{BÍn«sin(S-a)sin(g-6)sin(.S-c)} =cn/cn(1_2Ä)jSill t>-C COS ttC

so that

(6) cos CN, CN'= ^£^[dn(l -2A)±Acn(l-2A)]=sn2/P'dn(i±A)P,License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

534 A. g. greenhill: the elliptic integral

and

(7) cos CA^cos CN' = c' sn2 2/P', a constant,

a fundamental property of the sphero-conic.

71. If d^4 denotes the element of plane area cut out by consecutive quartic

curves,

(8) dA = c en w dn w en w dn w ■ F- dh-F' df

and if dw denotes the corresponding conical angle,

... ., . T , ., . ^'cnwdnucnw'dn«)' „„, ,„„(9) dw = 4 cos mtidA = 4 ,A , ,-*-.— FF'dfdh,w * (1 +csnwsnu») *

and putting w,w'= \fK' ± A7T?, this becomes by the quadric transformation,

l-A(dn47o-A2)(dn47o'-A2)

,t^ . IT* (1-A2)2dn27(;dn2-H/ , ,. ,„__,,-.dK= 4 —-/2V-'--;-■ — \(1 + kfKK'dfdh

' A' smccnwsn«)'cn«)'\!1+-

(- dn w dn w

- 2 (dn^-A2)(dn^'-A2) Kirdf,h(10) = i—p y\--,—r—p*-;-tt2 kk d/d/lK ' L — K ( dn w dn w + k sn w en w sn w en w )

= 2 ^^+^)àn2(W-w')-k2

dn'(w — w) J

= 2[dn2fK'-J^^KK'dfdh,

in which the variables are separated.

Performing the integrations between limits such that A ranges from 0 to 1,

and f from f to 1 and doubling for the whole conical angle il,

(11) ffdn2fK'KK'dfdh = P[(l -f)H' -zn/ZT] ,

ÎÎd^hKiKK'dfdh=SS{-1 "dn2(1 " 2h)K] KK'dfdh

(12) =(1-/)P' f [l-dn2(l-2A)P]PdAJo

= (l-f)K'(K-H),so that

(13) il=4(l-/)(A77' + P'P-PA')-4Pz»/P' = 27r(l-/')-4A"zn/A"',

as before, for the complete cone ; but (10) shows that the apparent area of any

curvilinear quadrilateral, bounded by two pairs of HolzmÜller's curves can be

expressed in the same manner by the I. and II. E. I.

Looking back over the calculation we conclude that the Stokes function has

the advantage in simplicity over the potential function, except for the case of

the circular ring of § 44.

1 Staple Inn, London, W. C,

January 14, 1907.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

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