THE RATIO BETWEEN THE HORIZONTAL AND THE.·VERTICAL' ,ELECTRIC FIELD OF A' VERTICAL

ANTENNA OF INFINITESIMAL LENGTHSITUATED ABOVE' A PLANE EARTH

by K: F. NIES SEN 538.566 : 621.396.672... Summary

We consider' a' vertical mathematical dipole in a height z« above theearth and calculate the ratio of the .horizontal to the vertical electricforce' as a' function of Ze, especially for directions making smallangles With the horizontal plane. It is shown that this"question is

, ,'. of interest in connection with the landing of an aircraft hy meansof a radio beacón. . . . .

Introduction. 'An aircraft landing in thick weather is assisted by a radiobeacon, which indicates in a certain way the line along which the landing·may best be executed. That line is usually given in two steps. First a;.vertical plane is indicated by.some means and then the line itself has to be ,· found in that plane bya second principle. .The first part of this "direction finding" -may be established by means oftwo equally strong vertièal aerials, which are in coun:terphase -and atthe same height above the ground. As soon as no. signal is heard the air-craft is in the' central plane of the emitters. The aerial on the aircraftis supposed to he a vertical one, but since the aeroplane is' often' ,more or less tilted about its major axis (pointing in the direction offlight), .horizontal electric forces will be received too. These horizontal com-ponents, caused by the two different emitters, having' different directions-do not cance1 each other, even not in the central plane, so that they mayprevent a pilot from finding the true central plane. Although practicalradio beacon.s are constructed in' a more ingeneous way, it still remains-

- desirable to diminish the horizontal electrical field components, Therefore'in this article attention is paid to these components, especially for small -angles of elevation, since a 'landing ..always takes place ·at an.angle of 4°· .. ~or 5° with the horizon. . <

, "

We use cylindrical coordinates r; i, cp with the origin 'on the surface oftheearth, so that the coordinates of the antenna are r = 0, Z = Ze. Then all .-'electric and magnetic forces at a point r;, z, cp can _becalculated from a .Hertzian vector function ~II(r, z), 'net contairiing cp. Therefore we may- 'just as well choose as an arbitrary point the point r, z, cp - o.

The vector function !1(r, z) will of course contain Ze as a parameter,' andcould therefore have been indicated by JI(·.l (r, ,z), which, however, willnot be done. Only for _:thecase z. = 0 we shall use the symbol HO (r, z).

The image of the emitter has the coordinates:' ,,-,

r = 0, z = -z •.

"

52 K, F, NIESSEN

The distancesof 'thè observer at r, z to the emitter and its image are:

s,= Vr2 + (Z-'Z.)2,

R2 = vr2 + (z + Z.)2and that between the observer and the origin :,'

R = vr2 + Z2,:Il (r, z) will depend on the dipole moment M and further, as already statedabove, on z. ~s a parameter, ' '. ~In gêneral we have 1):

, eik,R,' eik,R,Il(r,z) = M ~ -M Ir + IlO (r, z'+ Ze),

1 2where j = t 1, kl - 2n/A, ibeing the wavelength in vacuo. The',case ofan emitter above the earth can therefore be derived from that of an emitteron the ground by only changing the point of observation in the way indic-ated. This relation gives for z. = 0 (and therefore RI = R2) the identity

, (2)

~s one ~ould expect., If the observer comes nearer and nearer to the dipole [i.e. if we take,r -;. 0, z-;,z. and thereforeR1-;' 0,R2 -;.2z. andIlo (r, z+ Ze) ''';~1Io(0,2z.)],the relation gives ' , ' , . . .' ,

. ' eik,R, ; .Il(r, z) -;. M N'

" ,I

Il (r, z) ~ Il" (r, z),

, .which may be considered as a second control. For an ohserver on theground (z = 0) we nave RI 'R2 and therefore". .. '

Il(r, z = 0) -4,Ilo (r, Zo),

or in other words: .Upon investigating the Hertzian vector function on the' ground in' thecase of an emitter situated Zo above the ground, we find the same functionof-rthat would be found át points' at a height z. above the ground, using

.' an emitter on the ground. In this article we are interested especially in, observations above the ground, as is evident from the introduction.'

Favourable for the calculation is the-circumstance that for the usual wave-length the earth is a much denser medium th!ln the air. As a matterof fact: the'. electrical conductance 0'2 and the dielectric constant 82 óf,the, earth are so large that the value '

. . ~ .

- k 2 _.:c_ 82 CO2 + ,jw 0'2 (1)

• 2 - c21

(c = velöeity of light, w = ,2n/À = 2n'IJ: 'IJ= frequency) is much larger(in ahsolute value) than the cgrresponding value

for the air." ....

"

RA'rIO BETWEEN H\)RIZONTAL AND VERTICAL FIELD OF ANTENNA 53

In this case byusing a dipole onthe ground (z.== O),_Sommerfeld showedthat the Hertzian vector function for an' observer with ... . ,

is:., 'k R ,. iea '

tr (r, z) .. 2 M e~ [1 + 2 'Yeo (r) ,e-,e'(r)l_el' ~t]. : . . v~~

with

eo (r) = j2k(;, e' (~) , eo (r) (1 +~:;t

Van der Pol and Nie s s en ") generalized this formula for an arbitraryelevation zlr of the direction to the observer and found (again thanks tothe restrietion I k2~ I ~ kI2): " .

. eik,R __ ._ • iea, Ilo (r, z) = 2'!JfR [1 + 2 'Yeo (R) e-I!'<R>J!:'2 dt] , , (3) ,. , ' .. ' , . Ve'(R),

where '

eo (R) = jkl3

R2k2

e,' (R) ~ :, (~) (1 ; ~~)'. -In:the case of a radio beacon there is still another simplifying circiinistaxic~~the distance between ,observer and emitter is very large with respectto it (it <:'>.10 metres). ,,: ,, .The above formulae, being valid only for- R ~ it, can be expandedin negative powers of the ~stanée, and this expansion may be stoppedthe sooner the larger the' distance R. Especially for very large distanceswe find from (3), in the case' of an arbitrary elevatio~ zIt of the sightlille:

'17" [r, z) 2 M ,,~Rkl~\z~-<e, (R) (~+k, z)'+ ··,-1 . (4)kl R ". ". '

Even the chief term of (4) : ..' . eik,R k zIJO (r, z) = 2 M If' kIR + k

2z (5)

is sufficient, if'

2 eo- i.e. if

1 k2lkl H"

jk! R (1+(k2/kl) sin a)2 ~ sin a,.where sin a.= zlR~

In (5), just as in the above approximations, the variable z appears explicitly,_ but not r. The latter appears only implicitly by means of R . fr2 + Z2.

(6)

54 K. F. NIESSEN/

. ,allo _ 'k '11* .r .ar -J 1 R"In an analogous manner we find:

oJlO _ 'k 'll* ~ + (Oll*). OZ - J 1 R' OZ it

.'

This can he expresse~ hy writing. ,, ejk,R k z

. Ilo (r, z) = ll* (R,. z) = 2M R kIR -+ .k'(.z (:), I

where, öf course a new symhol, ll*, has to he int~oduced. Its meaning ishest seen from the right-hand memher of (7). Therefore we have: .'

allo _ (Oll*) r-ar- aR R

allo = (Oll*) ..~ + (Oll*) ,oz aR • R OZ R ,_

It is easy to calculate the required differential quotient

(Oll*) = 'k ll* +2M ik,~ ~ s2..- ,k2z, aR " J 1 e aR ?R kIR +

- on account of t~e very large distance R ~ .it, i:e.- 2nR

k]~. T~l.For then we have:

(Oll*) ..... . -', ejk,R 1" .aR • Jklll*+atermofthe\orderof2M R2 =R[jklRII*+ate~<=,>ll.~ .

.Thanks 1:9kIR ~ 1we may write:. Oll*·.·"

oR' = Jkill*,and eonsequently

We need these differential quotients in the calculation of the radial electricforce Er and the vertical- E. from the Hertzian vector function hy meansof well known formulae. . ,. ,

In, general we have (also for Zo >.0):.'. o2ll

Er='--oroz. o2ll

E.= kI2ll+-,. . OZ2

{8}, ..

{9}Thus in the case where z. = 0, using in. , o2ll0

Er (ze'= O) == oroz

the approximation for llo me~tio~ed anove; ~e have:

Er (~~. ' 0) = ~jk! k]R kt2' k2z' 2 M ejk,R [i;: + kIR ~ ~2Z] (10)

1E 1- 2M.. kl 1k2, I· 'cos a I":;; +" . 1 +'k pi

' T - If k • ] sin a k lZ.'. 11+k2sinal ,klR(I+~sina)'

1 kl

(ll)

RATIO BETWEEN HORIZONTAL AND VERTICAL FIELD OF ANTENNA 55

-For an ordinary angle of elevation (z/R C"..:l 1) we may only use the firstterm of [ ], but in the case of a very small angle of elevation both termsof [ 1 are to be taken into account. From its derivation we see that for-mula (10) may he used as soon as (7) is valid. This latter formula isoften called the reflection formula, for besides a direct wave it containsanother radiation which can be considered to be reflected against thesurface of the earth. For, (7) may be' written: .

'no (r z) = Meik,R [ 1 + k2z - klR) = M eik,R [ 1+ ILsin a-I] -', '. R klR + k2z ,R ILsin a + 1 '

with

,.' zsma=-. R

ILsina'-1 ' ,-'ILsin a +.1 is the 'well known reflection coefficient ofFresnel forpl~ne

- wav~s with an electric vector ~nthe plane of incidence. ' . - 'For the .derivation of (7) we needed, besidès kl R ~ 1 and Ik221~ki2, also

the restrietion (6). , -Being interested only in the amplitudes of the' electric forces, we write:

- . -We have added here already a term klz.P which in the case we .are treating(z.=0) has to be omitted, but which wewill need afterwards in the case wherez.>Q. The explicit formula for P will therefore be given later ..

Formula (ll), without the term klzsP, has already been given by Bouma. '_ The method of calculating ET was originally indicated by Sommerfeldbut nobody was then interested in the relative small value of the horizontalelectric force at points on the ground. ' ' ' ,- ,-

Formula (ll) will be better adapted to practical application if weintroduce the modulus n and the argument X/2 of t~e index of'refraction

/ k 2 ' ;.",~ .; n2 eit:.k

12 -

1 k 2 1 'As k:2 ~ 1, we have n2

~ 1. _

, ' For 0'2 = 0 t;_heformulae (1) and (2) would here give 82= n2•

In general we now have: ' , . '

1E'I- 2M ' kl2 n cos aT _- X

, - R -v 1 + 2 n cos i ~in a + n2 si:à2 a'

'V [ -' k 21R2- : isin3 a siri.i '1 '

X . sin4 a+ 1 1 + 1C1 z. Q . (12), ' - 1 + 2 n cosisin a + n2 8in2 a ~,' -

,11

1 -

56 K. F. NIES SEN

For the case where z. > 0 we shalllater give the expression for Q.., The line from the radio heacon to a landing aeroplane having an angleof eleyation a, which is usually small (a ~ 1), both terms in [ ] are to hetaken into account. ' -;

Since the emitters ofthe heacon do not stand immediately on the ground, and ultra-short waves are used, k1 z. c-o 1 or more; so that we. need also.the term k1 z. Q. .' , ' . ~_,The purpose of this article is not only to derive the formula for Q hut also

to investigate the range of the variahles !l,-R, Ze, in which formula (12)may he used; hut first 'the range of formula (12) will he invéstigated forthe case where Ze = O. The range of this formula is then the same as thatfor (6). , "

Considering the modulus of the Ieft memher of (6), and by-making useof n and x, we ohtain: ' . ' .

,'. ~n

~z~ •

1+ 2 ncos ï sin ,a+ 'n2 sin2 a ,

The required height of the point of observation depends therefore on theelevation a and also on n and X, i.e, the range of (12) is altered when thephysical properties of the soil are altered.

The range is also included in:2nR = Rf::;"'_ . ,'. n, , "

À. • ~" ,', , •, ' 'sina(1+2neosîsina+n2sin2a)

wher~ R' ,=' 2n X the distance from the ohserver to the emitter, expressedin the wavelength as a unit. , ' , 'As we are only interestad here in 'the order of the values, the true value

, , ,of cos "(./2(lying hetween Land 1/2 l'2) does not interest us and the 'rangewill depend chiefly hy means of n on 'the soil.' We, take simply cos X!2 === 1and siua = a. . "The last value is right only for a ~ 1, hut for aNI we still have 'sin aNI(e.g. a = n/4 gives sin a = 1/2 l'2),. so'as to the question of order we still .may set sin a = a. '. . , '

We shall now introduce a small quantity u hy means of e- \I

v • ,

for we had n2 ~ 1., ,_n heing of thé order of U-I (n N U-I), we see from the last equation that

'the reflection formula may he used as soon as '. '." , ,,-1

, a (1 + ,,-1 a + ,,-2 a2) ~ Rf:• I

The order of the left memher is indicated: ,in the case where a ~ u hy the third term, i.e. a~-2a2,

,in the case where a co x hy all three terms, i.e. " or a, in the case. where à ~ u hy the" first term, i.e. a.In orde~ that the reflection-formula shall ,he permitted it is thereforeneccssmry to choose the order of R' according. to the order of smallnessof the angle of elevation.

/'

RATIO BETWEEN HORIZONTAL AND VERTICAL FIELD OF ANTENNA 57

As a matter of fact we h~~e to choose:,

for a ~ u,UR/~,P".-;;_äU-I

'R"~-,.' a

for a e-c u,

for a ~ u, R/~a

Now for a small angle a we have

RI. " R' 2n R sin a 2n z Ia =sma' = --zÄ -T-/

. introducing a symbol z' analogous to R'. "For a C'>o.) 1 we have a > sin a,' but a N sin a still, holds .:We therefore

may always ~ite: 'Z' c-o aR',

so that for z' we obtain the ranger-

in. the case where a ~ u

in the case where a C'l u

in the c~se where a ~ u

r. u:t' ~ -2;: az' ~ U-I

Z' ~ U-I. ,

, ,and in general:

(1~)

Suppose ,ve are concerned with a ~oilwhose electric constauts are known, I

I.e. we know u. We let the angle a take various values and each time wetake a point P on the line, of sight whose (vertical) ordinate z Is givenby Af2n X the right member of (13). '

We then obtain a series of points forming a curve-C.

/ .

Fig. 1. Diagram for the validity of the reflection formula.

Above that curve C those-points of observation lay where the reflectionformula is applicable, since for them the inequality' (13) must be valid. Buton the curve C and below it', is not permitted to use the reflection formula.

\ Thus we see for instance that the reflection formula is never applicable tothe surface of the earth, not even at very large distances from the e~itter.

58 K. F. NIESSEN

,In order to state' where the reflection formula may be used we have todefine numerically the condition for a quantity A being large with respect toanother quantity B (A ~ B), since, accordingly to (13) the reflectionformula requires that: . ' ,

,it ' ,,-1Z ~. 2'.Tt • 1 :t- ,,-1 a + ,,-2 a2 •

N(lW we have already assumed that Ik221/k12~ 1.As may be seenfrom (1)and (2) this quantity depends on it and on the constants of the soil.- The usual wavelength for aeroplanes being it ,9 m, we give below the

value for it C'-' 10 metres:for sea water k22/k12 = 80 + 600 j, k22/k12 _ n2 <SJ 252,for fresh water "22/k12 = 80 + 0.6 j, k22/k12 _ n2 <SJ 92,for damp soil k22/k12 = 10 + 3 j, 1£22/1£12 n2 <SJ' 32,for dry soil . k22/k12 = 4 +0.06 j, k22 /k12 = n2 <SJ 22•

We therefore shall call A ~ B: when at least, A CS> 10 B. The inequality(~3) ,may then be written as follows:

for a <SJ,10" i <SJ'itf2'.Tt • 10 ,,/a2 or more,for a. <SJ" . : Z <SJ itf2'.Tt • 10 ,,-1 or more, 'for a <SJ,,/10 Z <SJ'itf2'.Tt • 10 ,,~I or more.

The smallest height of a point of observation at which the reflection formulais applicable is therefore:

.1

it . ,_. 10,,-1.2'.Tt .

But at the same time we must have an angle of elèvation a <SJ" or. a'~"., . .Since for a landing aeroplane the angle, a will never be larger than 40 or ,50 we have for this angle, exp~essed in radians 1)1

1a < n:

For dry' and for damp soil (the cases which. occur most commonly' fora radio beacon) we have '

. 1 1,,= <SJ'n i2 10'"

, .

so that of the above three ranges of a only that indicated by a co " willinterest us directly. For such an' angle of elevation of the sightline the'height, of the landing aeroplane above the ground. may be tiot less than

2it ·10 ,,-1 =,~ n y2 = 2 nit,'.Tt ' '.Tt'. -

ifthe reflection. formula is to be used. ,We should however not forget. that this. holds only for the case where z, = 0,

where the' emitter stands immediately on the ground. ,We 'shall now revert to the case where z. > O.

First we shall investigate the range of applicability for the reflection-

1) Sin~ewe used sin ~ = a, when a <: 1.

-;

RATIO BETWEEN HORIZONTAL ,AND VERTICAL F.IELD OF ANTENNA 59

formula imd subsequently thè fórmula for Q in" (12) will be derived.For an emitter above the ground (zo> 0) we have exactly (as already,stated in the beginning) : .'

, , . ' . M eik,R, eik,R, - ~Il (T, z) = ,Rl " M R

2' + Ilo (T, Z + ~o) (14)

As to the last term we are concerned, so to speak, with an emitter on theground and an ob~erver "at the height Z + Zo, and from the abo~e we knowthat for a small value of the angle ,,' \, '

Z+ Ze 'are sin r=======~==~y T2 + (z + ,Zo)2

" (being, in this altered problem, the angle of elevation) the reflection formulamay be used for calculating of Ilo (T, Z + Ze) as soon as .

'z + Zs > 2 n À,assuming

, ~., .: eik,R. k2 (z + 'Ze)/ ' IJ (T, Z + Ze) ,- 2 M R

2kl R

2+ k2 (z + Zo)' (15)

...Here again R2 appears, since the dist:ance from the altered point of ~bser-.vation (PI in fig. 2) to an emitter on the ground is exactly the distance fromthe realobserver to the image of the real emitter.

, p'

, ", Fig. 2. Notation of. lines to emitter and -its image., ,-Substituting (15) in (14r we obtain:

u (T, z),=M eik,R, [1+ eik,(R,-R,) k2 (z + Ze) - kl R2]. (16), ' Rl k2 (z + Ze) + kl !12

Th~s formula (16) may be called' the reflection formulá for the ease w'her';, z.> 0, and according to its derivation the range in which it holds good,for _small angles of elevation of the sightline is. given by .

Z + z.~ 2 n À.

We shall now de~ive Q' in (12). . _RIÀ being large and zei À being "-' 1, the three lines starting from Pand'indicated by RI' Rand R2 may be considered parallel to one another, ROthat - '. . "

,,' - .' ,- z.zR2 -;-- Rl = 2 z. sin a , 2 -R-'

Substitution ID (16) gives there a power of e which may be. expandedas follows:

60 K. ],'. NIESSEN

1+ .2kIz.z+t JR····

if

This restrietion is equivalent to4nzo À-.11.-N 1 (or z. N 12)'

since for a landing aeroplane Wehad alreadyZR~l.

Now for a usual wave beacon we have Za N .11./2,so that the restrietionz. N }"/12 is' not fullfilled and in the above expansion we ought to takemore terms into account. Nevertheless, in: order to avoid too compli-cated formulae we neglect these other terms and will be content to in-vestigate the primary effect of raising the emitter a little above the.ground. . .

Substitution of the eXJIansion in (16) gives:

IJ (r' z) = M eik,.R [ 2 k2 (z + zo) ,_ . 2 kl z. Z+' . . R k2 (z + Ze) + klR JR'

~ . 4 kl ZeZ . k2 (z +Ze) ]J R: k2 (z_+ Ze) + kIR'

In the combination Z + Zo the parameter Zomay be neglected, and fromthis rather simplified' IJ-function we derive, by. means of (8) and the res-strictions already mentioned above, a formula for Er ofthe form of(11),where _. " I ,

.'p .kl . 2 '+ . . 2' 1 . kl I, '. (17)= - sm a J sm,« cos a - J - ---.--,,,--=,k2 ,k2Z+klR k2k2Z+klR .

For practical use it is convenient to again introduce n and X and to calculatethe modulus of Er. We then find formula (12), where (thanks again to therestrictions) :. .. '.' " .

Q. 2. X • 4' 2~' . siÄ X/2' " . ,.-- .+=- - sm - sm a - - ::--;~;;--;::--:----;::-'--'~ê::--:-------;;-~~

:', .. n 2 nkI2R2(1+2ncosx/2sina+n2sin2a)+ 2 sin3 a (cos2 a + ncos X/2 sin a cos2 a + 2 sin2 X/2) (18)'

kIR (1'+ 2 ncos xi2 sin a + n2sin2a) ':.~In the derivation of Sommerfeld's formula mentioned in the beginningas also in that of our formula (3), terms with l/kt2: R2 were neglectedcompared with 1, but here in (18) they were to be taken into account andthey' appear together in (12) with only small terms such as sin4 ei:

I~ order to compare the horizontal electric force with the vertical one wenow calculate the latter (Ez) directly for the case where Ze :> 0.' '.

Subatitution of (16) into (9) gives immediately Ez in the-range where thereflection formul,a may be used:

.." '

RATIO BETWEEN HORIZONTAL AND VERTICAL FIELD OF ANTENNA 61

k2 ''kR' -k sm a. . eJ 1 , 1

E== 2 M kl2 cos" a 1{. k X1+ k2 sin a

r " , »; cos2 ~ . 1' ]X 1+ kl z. (-J -k +-,- 'k + k R) ,

L " 2 sm a 2 Z 1(19)

so. that

~ith

(20)

.. P _ .kl, ' 2' . + kl/k2 - sin a cos2 aI-Jk2 s:n a klR+k2Z'

P " kl + cos a: cotg a2_:_-J-

'. k2 ..kl R + k2 Z

Considering-only small angles a we find for the ratio "ofthe squares of the'moduli "

(21)

with',' A = 1 + 2 ncos~/2,s~a + n2 sin2 a (22)

B ='_:_,2 sin '1../2 " (23), nA k12 Z22 cos2 a ," '

C = 'A~lZ (1 ~ n sin a cos X/2) . (24),

B being negative and C positive we see that the effect of raising-thedipole a little above the ground will be a' diminution ,of the' horizontalelectrical component with respect to the' vertical one, especially in therange of small angles of elevation. The extent' of this effect can be 'calcu-latedby means of (21).' ', "

In the introdu~tion' we have already observed that theexistence 'of ahorizontal electric force is very troublesome for the landing, the avigationbeing based entirely upon; the exclusive reception of the vertical component,a condition which is not 'fullfilled when the aeroplane is tilted around itshorizontal axis (lying in the line of flight h as 'often happens. ' ,

Fig, 3." A radio beacon consisting of three vertical' dipoles,

.r:

K. F. NIESSEN

We therefore try to find a way of diminishing the ratio IErl : IE.I.'Aswe have already seen, one way will be to raise the aerial a littleabove the ground. ,Now a radio beacon does not contain one aerial only. A well-known

type comprises three vertical aerials, one right between the other two.The two outside aerials (A' and B in fig. 3) differ 1800 in phase withone another and the central one (0) 900 with each ofthe others. A and Bare of equal strength and -since the avigation is based upon receivingboth equally, we shall take as a point of observation a point P /inthe central plane. . , . .

Each of the emitters gives at P a horizontal electric force, which may becalculated from the Er-formula (10) with

M=MA=MB •.

These forces are differently directed, since that ofA lies in the vertical planethrough PA and the other in the vertical plane through PB. Moreover theemitters A and B are in opposite phase, so that, if the horizontal component

_of A 'points from P' towards A' (fig. 4), that of B is directed from B' to P',

.~.I ,. .I' ,

0'1 do' PI, , ~~1==iL----" . . I',JI E'I' r,S

A''1SJ18

, Fig. 4. Effect of horizontal electric forces in a radio beacon (seen from above).

so that a horizontal electric force ElJI results in the direction perpendicularto P'0', the modulus <;lfwhich is: .

Ie; I= 2 I Er I . sin ~:I , _ •

Here we have to substitute for IEr I the formula (12) with M = MA =MB.'and ~ is an angle which increases upon approaching the beacon.,ElJI is the component that spoils the avigation if the .aerial is .tumed alittle sidewards in a vertical plane perpendicular to P'O'" the horizontalcomponent of the emitterf) being'not troublesome when the aircraftisalready flying 'towards the beacon. If this direction is not yet found thehorizontal forces of all three emitters come into play. . ' ,

From this it will be clear that diminishing the ratio IEr I : Ie; I for eachaerial is always- favourable for the avigation. Ari.other question is i~ howfar might the real aerials be replaced liy mathematical dipoles in aproblem like the one ,considered.

Eindhoven, April 1945.

REFERENCES

1) For the derivation ofthis formula, stated for the ~st time by Sommerfeld, we referto K.F. Niessen, Ann. der Physik18, 89~, 1933. .' .

2) B. van d er Pol and K. F. Niessen, Ann. der Physik 10,485, lQ31, formula (19).