A Method of Reduction of a Quartic Surface Possessing a Nodal Conic to a Canonical Form.With an Application of the Same Method to the Reduction of a Binodal Quartic Curve to aCanonical FormAuthor(s): John FraserSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 24 (1902 - 1904), pp. 71-84Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20490577 .

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[ 71 ]

X.

A METHOD OF REDUCTION OF A QUARTIC SURFACE POS

SESSING A NODAL CONIC TO A CANONICAL FORM.

WITH AN APPLICATION OF THE SAME METHOD TO

THE REDUCTION OF A BINODAL QUARTIC CURVE

TO A CANONICAL FORM.

By JOHN FRASER, M.A., F.T.C.D.

Read DECEMBER 14, lt03. Published JANUARY 26, 1904.

THE equation of a quartic surface possessing a nodal conic may be written in the form

[ae2 *+ fy2 + yzflt ? w2 {[a, , , d, , g h, 1 m n] [xyzw] } = 0

We may write x for xv/a, &c.; and then the equation becomes

[X2 + y2 + z2]2 + W2 { [abedfyk in] [xysw]2} = 0.

If the quadric

Xe + y2 + Z2 + [ax + fy + yz + 8W]w = 0

has double contact with the quartic, then it must have double contact

with the quadric

[ax + fly + 7Z + SW]2 + [abedfgk lmn] [xyzw]2 = 0;

and hence

[aX + fly + 7Z + SW]2 + [abedfgk Imn] [xYzw]2 + 2[x2 + y2 + s2 + w(ax + fly +7y + Sw)] -Li,

where L- 0 and Xf= 0 are two planes. Hence, since every plane meets this quadric in a pair of lines,

every first minor of the discriminating determinant of this quadric must vanish.

Let A denote this determinant: then

A= a+2X+a2 h+afl g+ay l+ a(X+S;)

h + fla b + 2X +f l2 f + fy m +8(A + 8)

g? + ya f + e- C + 2X + y2 n + y (X + 8)

I+a(X+8) m+,8(X+8) n+ y(X+8) c?(X+8)-_X2

X2

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72 Proceedinags of the Royal Irish Academy.

A 1 0 0 0 0

a a+ 2X+a2 h + aP g +ay + I+

a(X + 8)

h?A 8a b 3 +2A+ /32 f +/y rn + (X + 8)

y. g?ur+ya f +y/3 G-i +2X+ 72 n+y(A+8)

x+8 l+a+) + m+13(A+?8) n+y(A+S) d? (X S 8)2 _ X2

_ 1 -a - / - By (X + 8)

a A 2 h g Z

/3 h b + 2X f m

y q f c+ 2X n

X+8 I m n d - X2

Now since any first minor of the original form of A vanishes, then any first minor of the latter form muist vanish. Hence,

a+ 2X h 9 1

h b + 2X f m

gq f c+2X n

I m n d - X2

which is a quintic for A.

ii. aL +f,8X+ yN- (X ?8)D - 0,

where L, M, N, I) are first minors of i.

iii. -1 a / y

a a+2 h g

,B h b + 2X f

y g f e+2X

Since aL +/3M + yN+ (X + -S)D 0,

8D = - [aL + ,fM yN+ AXD],

and X2 + y2 + Z2 + W[ax +y + 4 z + 8w] = 0;

- (x2?+ y2 + z2) + a-Dwx + ?/Dwy + yDwz= - 8Dw2 = w2[aL + pff?+ yN+ A-D];

a[Dwx- w2L] + /[Dwy - w2ff] + y[Dwz- w2N] + D[x2 + y + Z2] - ADw2 = 0.

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FRASER-Reduction qf a Quartic Sufiace to a Canonical Form. 73

This denotes then a system of quadrics possessing a Jacobian surface.

That is, the system of quadrics passing through the nodal conic and touching the quartic surface twice possesses a Jacobian surface, anid there are five such systems.

Dw 0 0 ]xx- 2wL

1= 0 DO w 0 Dy 2w[ 0 0 Diw Dz -2wN

DX Dy Ps -XDw

= - D3W2[De + y2 + sZ) - 2w[Lx + My + Az] + DwD2].

J contains W2 as a factor; and the remaining surface is the quadric

e + y2 + 2 2w[Lx +-My

+ AN] + Xw2 - O.

D

And there are five, and only five, such quadrics. Consider the point which has w for its polar plane with respect to

au + ,8v + yw + T = 0,

where U = Dwx - w2L

V = Dwy - OM,

IV = Dies - w2N,

T = D [X2 + y2 + ze - Xwl

D[aw + 2x] = 0, D[f3w + 2y] = 0, D[aw + 2z] = O,

x, y, z, w being the coordinates of this point; hence if, in iii., we substitute for a, /, y,

x y

respectively, we get

to2 x y z

x a+2X h g =0 ,

y h b+2X f a g f c 2X

as the condition which the coordinates must fulfil-that is, the locus of the point is a quadric.

J= + y2 s2 - w (L,x + 1y + N1z) + Xw}

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74 Proceedings of the Royai Irish Academy.

where f(X1) = 0,

and f (Xj) a + 2A1 A 9 1

A b +J 21 f m

g f c + 2k n

Z m n d - kiz

Consider

f5 (r 2 + y2 +52j (LtZ+AfrY+Nrs)+Xr62],

Dr Ar3 + &C... . f (Xr)--8Ar5 + &0.;

hence the coefficient of (x2 + y2 + Z2)2 vanishes; and for the same reason the coefficient of (X2 + y2 + Z2) W [X, y, z] also vanishes.

The coefficient of

w2x2 = 2- 1 - -(A) + 4 . 1r

But since f (Ar) 0 ,

LrtA - r = 0.

Lr Ar =4Ar4 + &c.

Hence [2XrDr +4 = (16 -16) Xr4 + &C.;

therefore the coefficient of w2aP also vanishes. The coefficient of

2 = 6 -DrL r Lr9r I

Dr2f I (Ar) Dr)f (Ar)

but 1944- 1IT, D 0.

L44 = - 2h,\Q3 + &c....

Hence the coefficients of w2 [Xy, ya, zx] all vanish; and hence

Y5 f' A-D [Z2 - Yz -- 2w[Lx ? 4y + Y 4 A , D [2 + 2 + Z2 - +

___ __W2____

1f (Ar) LD

that is, that the squares of these five Jacobia-n Quadries are connected

by a linear relation.

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FRASER -Reduction of a Quartic Surface to a Canonical Loin, 75

Again, consider

j r[2 +.y2 + z2 - 2 (-Lp + XrY + Nr ) w + A2]Dr 8Ar3 + &C.,

Dr w+72 7=4+c

the coefficient of

(a2 + y2 + S2)2 a

> 8Xr4 + &C _.

f'(X,) the coefficient of

WX3 =>1_4rLr _ r[4X2+ &c*] = o; wx3 = 4f'(X7) = C4$fXr -0; hence the terms

w y[3Y, 3, 2y, X2 y2, y2z,; z2s, Z2y]

do not appear, as their coefficients all vanish for the same reason. The coefficient of

fX, [ Ix 2 + 4 _ [2Xr2r + 4XArr]

ADr - Lr2 =0

42Xr28X r3+ 4Xr2 (a f ) b + +) } + &c. 4Xr {4Xr + 4Xr (b +)rI1

a. 8Ar4l +.. = ~~~-a;

and of course the coefficient of w2y = - b, and of w2z2 = c. The coefficient of

w2xy 8Xrlrx) D 8Ar - 8XrI 8 11 = 8X4hh(c+2XA)(a-Vr2) +&c.] L(Ar) * f

(Xr) h. l

6Xr4 -_-...=-k f (Xr) 2h.

Similarly the coefficient of w2y2z - 2f, and of w2ZX - 2g. The coefficient of

WX= 154LAr Lr = - 41X75 + &c.;

hence this coefficient = - 21, that of w3y = - 2m, and of w'z - - 2n. The coefficient of

W IL r4m ll+m + nY + (d-X2) D = f(X);

L, H, N are only quadratics in A; hence

Q,3Dr = - XrfJ(A) + ArFlLr + mJfr + nNr, + d(-D] = d. 8Ar4 + &c.;

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76 Proeeedings of the Royal I}isl Academy.

, f(XD)

- 0, X y2 + tD - -d

-~ (X') -J2 A (x +[+52 + w2[(a, b, c, d, f,g, h, 1, m, n)(x, y, z,)fl.

Hence we have the equation of this quartic surface expressed in a canonical form

ViZ.: f'(-X) r '

where Pk) ' "

We might consikier particular cases of this quartic surface, accord ing as the conic becomes an ellipse, hyperbola, parabola, or circle.

In particular, the imaginary circle at infinity. In this case, the plane w becomes the plane at infinity, and the

axes x, y, z rectangular; hence we may pat, without introducing any other peculiarity, f = 0 = g= h, and a - = y 1 in the original equation.

And the Jacobian quadrics become of the form

2 (Lx + Ky + Nz) w X;2 + y2 + s22_s

+ AX2

viz., spheres, where

0 =f(X) a + 2X 0 0

o b + 2X 0 m

O 0 c+2X n

I m ti d -MX

The system of quadrics which have double contact with the surface must also be spheres, since they pass through the circle (imaginary) at infinity, and the locus of the pole of the plane at infinity with respect to them, that is, their centre, is

w2 x y

x a+2 0 0

y 0 b + 2A 0 -0,

z 0 0 e+ 2X

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FRASER -Reduction of a Quartic Su:face to a Cajnoical Form. 77

which shows that the five quadrics belong to the same confocal system.

_L2 +MI+ A+B C-X._D f(X) 2 = 2 -AN= BB

and hence, in this case, the identical linear relation becomes

Vs J, 2=O - o .

x2'y + z [(abcfgh) (xyZ)2] - 0

represents the general equation of a binomial quartic curve having the

points z = 0, x = 0; z= 0, y f Q or its two nodes.

z (ax + fly + yz) + xy = 0

denotes a conic passing through the same two points. If it has double contact with the quartic, then it mast also have

double contact with the conic

(aZ + fly + yZ)2 + (abefgh) (xys)' = 0; hence

(aX + fly + yZ)2 + (abebAh) (XyZ)2 i 2X [z ( + fy + ys) - xy]

and therefore every minor of the discriminating determinant must vanish.

Let A denote it.

4=3 a + a2 + X + afl g+ a(X+y)

h+ X s,8a b+/32 f+/3(X+?y)

g + (X + y)a f ? (X+ y) G(+ y)-X2

1 0 0 0

a a+ t hk + X + a, + a (X +y)

p h + ? + fla b + f2 f + f (A + y)

X?+ yg + a(X+y) f+f8(X+y) c+(X+ y)2-X2

_ 1 -a -f - (X+Y) a a h?X g

fi h + A b f

ly g f c - X2

and every minor of the latter form of A must vanish.

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78 Proceedhigs of the Royal Irish Academy.

a h?+ g

k+X b f =0;

g f c-X2

ii. aG +13F+ (X+y) C = 0;

iii. 1 - a - 13

a a h ? = 0.

,l h + A b

i. is a quartic for A, and to each value of A we have a linear rela tion connecting a, /3, y, given by ii., and a quadratic relation between a, ,3, given by iii.

xy s (ax + y -yz) 0,

aG+/F?+(X +y)C 0;

..a [ CZX GZ2] Z 3[Cy-F21 + Cxy + XCZ2 =

O

a[Cz-Czf+/8[Czy-Fz+ Cx+LY=

is an equation of enveloping conic; and its form shows that it belongs to a system which has a Jacobian cubic curve, viz.:

a 0 y

0 s x =0,

Cx-20z Cy-2Ez -2XCz

or - 2s[XG2+ Cxy-Fzx - Gzy] = 0.

Hence the system which, since it passes through two fixed points, has a common Jacobian conic

S xy -

[Fx + Gy]

+ As2

= 0,

a conic wbich also passes through the same two points. To each value of A, we have a corresponding conic

a (Czx - Gz') + ? ( Czy -F z) + CXy + X C2 0.

If this conic becomes a pair of lines, then

O C aC

C 0 3C =0,

aC 3C 2AC- 2aG -2/3F

- C2[2aflC -2AC + 2aG + 2F] = 0;

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FRASER-Reduction of a Quarlic Sw/ace to a Canonical Form. 79

rejecting the factor C2,

2aG + 2,BF- 2XC + 2a/3C = 0;

but

-1 a /a

a a h+A = 0.

8 hk+X b1

Eliminating / between these two equations, we get a quartic for a, and to each value of a one value, and one only, of ,B.

Hence, to each value of A, we have four conics of the system which reduce to a pair of lines, that is, a conic consisting of a tangent from each node of the binodal quartic.

The node of this conic, i.e. the point of intersection of these two tangents, must lie on the corresponding Jacobian. Hence, the four Jacobians, as written above, are the equations of four conics, each of

which passes through four of the sixteen points of intersection of the

tangents to the binodal quartic from its nodes; and hence, we may infer that the anharmonic ratio of the two pencils of tangents is the same, since the conics pass through the nodes.

If the point xyz has the line z = 0 for its polar with respect to

a [Czx - G21 + ,L[CZy _Fz2 + C [xy - As2] = 0,

then aCZ + Cy =

Q, aCZ + Cx =0;

hence S2 y

*

y a h+ X =0.

x h+X b

Hence, the locus of the poles of the line z = 0 with respect to the

conics of the system lies on a fixed conic; and the above is its equation determined in terms of the coefficients of the quartic.

Corresponding to each value of A, we have a conic. The Jacobian conics are

s FG+ FrY Xy 2+&c

Consider 4 Cr [y - t (Gx + FrY)

1~1 C'(X Y s]

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80 Proceedings of the Royal Irish Academy.

The coefficients of Xq2, zz2y, Zxy2 vanish since G is of the first order in A; Fis also of the first order, and C of the second order in A, f(A) being of the fourth order.

The coefficient of z4 is

Ar 20,

gOr ?r fF, + (c - x) Cr = fr(Ar),

IC2 = f (Xr) + gG, + fI =gGr + fFr, * f (Xr) - 0

* . the coefficient of z3 is zero.

The coefficient of

2zexy = :l (Xr G r_F& C,.

but

II G,CJ - I> =, 0 f(Al) O

14_ [4[ ArCr'Br 1 - 4X2 + &c.

f' (Ar) f' ~~(A-r) The coefficient of

=X z14 G 2

F2 BIC,r

- F;2 = Q Br- X2 + &C.; Ct2

Gr2 o

and

., EloftCr [*y _ g (G,z + F,y) + ASz2j t.

Again consider

<5> -C (12 + G,y) 2 2+

The coefficient of

= 4 ArC, = _14 ___ = - 1. x2Y2 (X,)~~f'(,.

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FRASER-Reduction of ai Quartic Surface to a Canonical Form. 81

The coefficient of

z2y = Y4 AriIPi - ,+.

f (Xr) - ( .

The coefficient of

$ r = = -a () C, g r (X) -r)

The coefficient of

= 4 r r

-;

G ) [r2C, f

(Xf' )

A* CGr+fr(A CfX)=0

iOr XjC,{cCrg G + fF,] =

The coefficient of

z3x = - 2>1zXrfir,rf'A) = -2 ** '2 =

2 ZZ = ~f'(Ar) Lf(Xr)J

The coefficient of

21, f5f'A,) -2V~1 .--=-2

C, ~~f'(Akr) f'(Ik) =-k

Hence

- PkCF )

(z+ 0rY)C

+ ArZ2

-g2y2 + z2[(a, b, c,f, g, h) (x, y, s)2],

and Z fTj) [ Er ; Y) ? Arz] 0.

The bicircular quartic might be treatedl in precisely the same manner, z = 0 denoting the line at infinity in the plane, and x + iy written for x, x - iy written for y. But it can also be treated directly, thus

(e2 + y2)2 + ax2 + by2 + c + 2gx + 2fy = 0

denotes its equation.

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82 Proceedings of the Royal Irish Academy.

If the circle e9 + yZ + 2az + 2/8y + y = 0 has double contact wit it, then

(2ax + 2fiy+ y)2 +ax'+ by2 eG 2gx+2fy+ c+ X[x2+ y2+2ax? 2/3y+y EEL2;

hence, as before, every minor of

A= a tXA + 4&x 4aj3 2a-y + g + aX

4fla b +X + 4f2 2f31y i f + ?X

2a7y-ig+Xa 2/3y+fiX/3 c ?Xy?y + i

- 1 4a - 4/ - (k + 2-y)

a +iA 0 9

/3 0 b - A f 0.

A A A2

i. f el O g f 4~~~~~~

1i. I1 4a - 4f

aa 6? / 0 0, or ? + ---+ I =0;

,B A b +k g- f f

4 2 9 r a A b+ A4

i. Shows that X is determined by a quartic.

ii. Shows that the centre of the enveloping circle moves on a 1xed

conic, and also to each value of X we have a determinate conic, and

the conics are con0ocal.

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FRASER-Reduction of a Quartic Surface to a Canonical Form. 83

iii. Shows that the circle cuts the fixed circle

X2+Y2+ x+^ 2fAy+2 ?k= a+xx +

orthogonally; but, since

a + A1 b + A, 4

g2 -+ - O

2q'2 2f2 _ A1 + A2 (a + X,)(a + A2) (b + A,)(b + A2) 2 '

hence the fixed circles are orthogonal, and as in the case of the binodal

(luartic the 16 points of intersection of the tangents from the circular points to the quartic lie by fours on these circles,

tf'-k) L(x2 + y2) + 2x + -+

where C= (a + A)(b + Xr);

and

Y4 I(a Ar) F( +2+I 2gx 2fy Xr 2

f'I(XI) L ~ aX b+Xr 2j (X2 + 2)' + ax + by" + 2gx + 2fy +

cy

where f(Xr) = a + Ar

a b + A, f =0.

gq f =

The reduction is just the same

C= (a + X)(b + A),

- F = g(b + A),

-G =f(a+)A),

&C. .

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84 Proceedings oJ the Royal Irish Academy.

It is interesting to show, from this point of view, that

f ' (X) R 2

(a + X)(b + X)

112= .9 + f2 A (a - A) (b+ A2 2'

+2 f2 2

-f(A.) a + A b + X 4 (a +X)(b + A)

also

.9 f 2 A 5 f(A) f' (A) (+ )2' (b+ X)2 2 A. (a + A)(b + A) (a + A.)(b +A)'

since

f(A.) = 0; hence

Jr

_,,>4 A. Jr22 a 41I 7J" (2 +j?22 +ax2 +by"2+2gx?+2fy-i-c.

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