400 
ON THE JACOBIAN SEXTÏC EQUATION. 
[776 
this becomes identical with the foregoing Tschirnhausen-transformation equation; thus 
ae + 3c 2 = - 9/3 + 368 + 128, = /3 - 9 
8 + 48; 
and similarly 
3a 2 e 2 - 2ac 2 e + 15c 4 = /3 2 + 243, 
/38-1872, 
8 2 + 3840. 
So for the constant term, + la 3 cf 2 gives the term 432/i in/8, and + la 3 e 3 , &c., give the 
remaining terms — 729/3 3 , &c. of the value in question. 
It only remains to verify the equality of the coefficients of X, 
216 a/A = VO or 46656A=D. 
Here □, the discriminant of the quintic (a, 0, c, 0, e, l) 5 , from the general 
form (see my Second Memoir on Quantics, [141], or Salmon’s Higher Algebra, third 
edition, p. 209) putting therein b = 0, d = 0, is 
□ = a 4 f 4 + 1, 
a 3 ce 2 f 2 + 160, 
a 3 e 5 + 256, 
a 2 c 3 ef 2 - 1440, 
a 2 c 2 e 4 — 2560, 
ac 5 f 2 + 3456, 
ac 4 e 3 + 6400, 
and writing for a, c, e, f their values 1, 2 \J8, 9 (—/3 + 48), 216/i, the value becomes 
□ = (216) 2 . h 2 . 
+ 4>S2h^/8 • 12960 (/3 - 48) 2 
+ 34560 (/3 - 48) 8 
+ 55296 8 2 
- 25 6 . 9 5 . (/3 - 48) 5 
- 10240 . 9 4 . (/3 — 48) 4 8 
- 102400.9 3 . (/3 - 48) 3 8 3 . 
The whole divides by (216) 2 , and we thus obtain 
A = h 2 + 2h a/8. 60 (- /3 + 48) 2 . + (- /3 - 48) 3 . 324 (/3 - 48) 2 
- 240 (-¡3 + 48)8 + 1440 (/3 - 48) 8 
+ 256 8 2 + 1600 S 2 , 
which is, in fact, equal to the foregoing value of A.