777] 
A SOLVABLE CASE OF THE QUINTIC EQUATION. 
403 
have the above-mentioned values, omitting therefrom the terms — -; we find without 
difficulty 
2 - BE - CD, 
a 
d! 
â' 
e' 
a' 
/'_ 
2 -, = — B 2 D - BCP - CE 2 - D-E, 
a 
-, = - B 3 G - B 2 E 2 + BODE + BD 3 + C 3 E + CW- - DE 2 , 
i 
- / = -B 5 + 5B 3 DE- oB-G 2 E - 5B 2 GD 2 + 5BC S D + 5BGE 3 
- 5BD 2 E°- -C 5 + 5CD 3 E - 5CD 2 E 2 - D 5 - E\ 
and hence, when E = §, we have 
2-! = - CD, 
a 
2 — B 2 D — BG 2 , 
a 
— = — B 3 G — BD 3 — C 2 D 2 , 
a 
Îl = -B 5 - 5B 2 CD 2 + 5 BG 3 D -G 5 - D 5 , 
or, as these may be written, 
-24 
a 
-2- 
” a' 
= CD, 
= B 2 D + BC 2 , 
_ <L-4i^ = B 3 C-BD 3 , 
f 
= B 5 + G 5 + D 5 -10 — (B 2 D - BG 2 ), 
equations which imply a single relation between the coefficients a, c, d, e, f. 
Supposing this satisfied, we may attend only to the first three equations; or, writing 
for convenience, 
?=- 2 7' 
a 
= — - (ac — b 2 ), 
= - 4 (a 2 d - 3abc + 26 s ), 
a 3 
e = - —, - 4 \, - - - 4 [a 2 (ae - 4bd + 3c 2 ) + (ac - 6 2 ) 2 }, 
a cl 2 a 
the equations are 
7 = CD, 
8 = B(BD + C 2 ), 
6 = B(D 3 - B 2 G). 
51—2