859] COMPLEX OF LINES WHICH MEET A UNICURSAL QUARTIC CURVE. 429 
If for the six coordinates we substitute their values, fiz — 7y, &c., we obtain 
n, = (x, y, z, w) A (a, /3, 7, 8) 4 = 0, which, regarded as an equation in (x, y, z, w), is 
the equation of the cone, vertex (a, /3, 7, 8), passing through the quartic curve; 
this equation should evidently be satisfied if only ®, P, Q, R are each = 0, viz. il 
must be a linear function of (©, P, Q, R); and by symmetry, it must be also a 
linear function of (® 0 , P 0 , Q 0 , R 0 ), where 
©o = aS — /3y, P 0 = a 2 y — /3 3 , Q 0 = ay 2 - ¡3*8, P 0 = 7 3 - /38*, 
viz. the form is il, = (©, P, Q, R) (©„, P 0 , Q 0 , P 0 ), an expression with coefficients which 
are of the first or second degree in (x, y, z, w) and also of the first or second degree 
in (a, /3, 7, 8). 
To work this out, I first arrange in powers and products of (a, 8), (/3, 7), ex 
pressing the quartic functions of (%, y, z, w) in terms of (©, P, Q, R), as follows : 
0= 
a 4 
-b 3 h 
+ bf 2 g 
+ c</> 
- acfh 
+ 2c 2 h 2 
- 4a 2 eh 
+ af 3 
- a 3 f 
a 4 
a :: 5 
- Z 4 
+ yzw 2 
0 
- z 4 + yziv 2 
- zR 
a 2 8 2 
- 2xyzw 
+ 2y 2 z 2 
- 2xyzw + 2y 2 z 2 
-2 yzQ 
ad 3 
+ x 2 yz 
- y 4 
+ x 2 yz - y 4 
+ yP 
5 4 
0 
a s /3 
- zw 3 
+ ziv 3 
0 
a 2 p5 
+ 2xzio 2 
+ //2 2 M; 
- 3xzw 2 
- xzw 2 + yzhv 
- zwQ 
a/35 2 
- .X 2 2iy 
+ 3//% 
- XlJZ 2 
- 4xyz 2 
+ 3x 2 zw 
+ 2 x 2 zw + 3 yhv — oxyz 2 
+ 2x20 - 3 yQ 
/35* 
+ .-r?/ 3 
- x s z 
+ xy 3 - x 3 z 
— xP 
a 3 y 
+ 2 3 ii> 
— yiv 3 
+ z 3 w-yw 3 
+ wR 
a 2 yb 
+ 3oS2 3 
- OT/M> 2 
- y 2 ziv 
- 4y 2 ziu 
+ 3 xyw 2 
+ 3xz 3 + 2xyw 2 - 5y 2 zw 
+ 2ijwQ + 3zQ 
ay8 2 
+ 2 x 2 yw 
+ xyz 2 
- 3 x 2 yw 
- x 2 yw 4- xyz 2 
yd 2 
- X 3 i/ 
+ x 3 y 
0 
a 2 /3 2 
0 
a 2 /3y 
+ £M> 3 
- yzw 2 
+ xw 3 - yzw 2 
+ w 2 Q 
a 2 y 2 
- 3xz 2 w 
+ y 2 w 2 
+ 2 y 2 w 2 
— 3 xz 2 w + 3 y 2 w 2 
- 3wQ 
a p 2 8 
- 3i/ 2 iu 2 
- xz 2 w 
+ 4 yz 3 
- 3yho 2 - xz 2 w + iyz ?> 
- 4z 2 Q + 3wQ 
apyd 
- 2a; 2 2t) 2 
+ 2xyzw 
+ 8xyzw 
- 8y 2 z 2 
- 2x 2 w 2 +10xyzw - 8y 2 z 2 
+ (- 2 xw + 8 yz) 0 
ay 2 5 
- 3x 2 z 2 
- X?/ 2 iy 
+ 4 y 3 z 
- 3x 2 z 2 - xyho + 4y s z 
- 4y 2 B - 3xQ 
/3 2 5 2 
-- 3xy 2 io 
+ a; 2 * 2 
+ 2 x 2 z 2 
— 3 xy 2 w + 3x 2 z 2 
+ 3x Q 
¡3yS* 
+ 
- xhyz 
+ X 3 W - X 2 IJZ 
+ .i ,2 0 
y 2 S 2 
0 
a(3 3 
+ ?/u> 3 
— z 3 w 
+ yw 3 - zho 
- wR 
a/3 2 y 
+ xzw 2 
- 4 yz 2 w 
+ 3yzhv 
+ xzw 2 - yz 2 w 
+ zwQ 
a @y : 
— xyw 2 
- 4xyiv 2 
+ 8y 2 zw 
- 3y 2 zw 
— 5 xyw 2 + oy 2 zio 
— 5 ywQ 
ay 3 
+ 3x 2 zw 
- 4//%e 
+ yho 
+ 3x 2 zw - 3yho 
+ 3 ivP 
/3 :! 5 
+ 3xyw 2 
— 4xz 3 
+ xz 3 
+ 3xyw 2 - 3 xz 3 
- 3 xR 
/3 2 y5 
- X 2 ZW 
- 4x 2 zw 
+ 8 xyz 2 
- 3 xyz 2 
- 5x 2 ziu + 5xyz 2 
- 5xzQ 
Py 2 d 
+ xhjio 
- 4xy 2 z 
+ 3 xy 2 z 
+ x 2 yw — xy 2 z 
+ xyQ 
y ! 5 
+ as 3 *: 
- xy * 
+ x 3 z-xy 3 
+ xP 
|3 4 
+ Z 4 
- aszo 3 
+ z 4 — xw 3 
+ zR-io 2 Q 
P 3 y 
- 4j/z 3 
+ 4xz 2 w 
- 4yz 3 -l- 4xz 2 w 
+ 42-0 
P*y 2 
+ 6?/ 2 2 2 
+ 2 x 2 w 2 
- 8xyzw 
+ 2x 2 w 2 - 8xyziv + 6y 2 z 2 
+ (2xw - Cyyz) 0 
py s 
- 4// 3 2 
+4xy 2 w 
— 4 y 3 z + 4 xy 2 w 
+ 4 y 2 Q 
7 4 
+ ?y 4 
- ,T% 
+ y 4 - x 3 w 
- yP-x 2 Q