436] 
ON A CERTAIN SEXTIC TORSE. 
107 
13. ihe equations, putting after the differentiations w — 0, and writing for shortness 
(*) in place of (*^x, y, zf, become 
tZ© e£© d® 1 1 i i 
dx dy ' dz + X V Z *) ~ (0 + a y • (0 + @y • (d + yf ’’ (0+ 8) 2 ' 
Now, observing that the second factor of © vanishes for the values 
a(0 + a) 3 , b(0 + ¡3) 3 , c(0 + y) 3 of (x, y, z), 
we have simply 
^ = (g 2 h 2 x + h 2 f 2 y + f 2 g 2 zf. 3a 2 [(a 2 x + b 2 y + c 2 z) 2 - 9b 2 c 2 yz]. 
But 
a 2 x + b 2 y 4- c 2 z = a 3 (9 + a) 3 + b 3 (0 + /3) 3 4- c 3 (6 + y) 3 , 
= 3«6c (0 + a)(0 + /3) (6 + 7), 
in virtue of the relation a (6 + a) 4- b (6 4- /3) + c (6 4- 7) = 0 and hence 
[(a 2 x + b 2 y + c 2 zf — 9b 2 c 2 yz] = 9b 2 c 2 (6 4- ¡3f (0 + y) 2 • [a 2 (0 + a) 2 — be (6 + ¡3) (6 + 7)], 
= 9b 2 c 2 (0 + /3) 2 (0 + y) 2 Q, 
where 
Q = a 2 (0 + a ) 2 — be (6 + /3) (6 + 7), 
= b 2 (6 4- /3) 2 — ca (6 4- 7) (0 4- a), 
= c 2 (0 + 7 ) 2 — ab (9 4- a) (0 4- /3). 
Hence 
^ = 27a 2 b 2 c 2 (g 2 h 2 x + h 2 f 2 y +f 2 g 2 z) 3 (0 + /3) 2 (9 + y) 2 Q, 
and similarly 
= 27a 2 b 2 c 2 (g 2 h 2 x + h 2 f 2 y + f 2 g 2 zf {9 + y) 2 (9 + ctf Q, 
dy 
~ = 27a 2 6 2 c 2 (g 2 h 2 x + h 2 f 2 y 4- f 2 g 2 z) 3 (0 + a) 2 (0 + /3) 2 Q; 
dz 
whence the above-mentioned conditions reduce themselves to the single condition 
(0 4- 8) 2 {34) 4- xyz (*)} = 27a 2 b 2 c 2 (g 2 h 2 x + h 2 f 2 y +f 2 g 2 zf (9 + a) 2 (0 + PY (0 + y) 2 Q- 
14. But we have 
g 2 h 2 x 4- h 2 f 2 y +f 2 g 2 z 
= g 2 h 2 a (0 + 8 +f) 3 + h 2 f 2 b (0 + 8 + gf +f 2 g 2 c (0 + 8 + Kf, 
= (0 + 8) 2 [(g 2 h 2 a + h 2 f 2 b 4- f 2 g 2 c) (0 + 8) + 3 (gha + hfb +fgc)fgh], 
= — abc (0 + 8) 2 [(gh + hf+fg) (0 + 8) + 3fgh], 
= - abc (0 + 8) 2 [gh (0 + a) + hf(0 + P) +fg (0 + y)], 
= — abc (0 + 8) 2 P, 
14—2