170 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
Article Nos. 293 to 295.—Hermite’s application of the foregoing results to the deter 
mination of the Character of the quintic equation. 
293. By considerations relating to the form 
|[Dj? — QBDtv — D (A — lOHB) v 2 ] + D [— Bu 2 + 2D x uw + 9BD — 10A Aw 2 ]j, 
M. Hermite obtains criteria for the character of the quintic equation f(x, 1) = 0. 
294. If D = — , the character is 3r + 2i, but if D = +, then expressing the fore 
going form as a sum of four squares affected with positive or negative coefficients, 
the character will be or or 2 + 4i, according as the coefficients are all positive, or 
are two positive and two negative. Whence, if i\T denote as above, then for 
D = +, N=— , A = + > B=—, character is or, 
Z> = + , N = ~, BD 1 = + ) 
and 
D = + , N = + j 
and further, the combination D = + , N = —, A = 
set of criteria). 
character is r + 4i ; 
—, B = + cannot arise (Hermite’s first 
295. Again, from the equivalent form 
^ |A (t 2 — Dv 2 + 2D aw - 10 Aw 2 ) + BD (lOAa 2 — Gtv — u 2 + 9Dw 2 )j, 
which, if co, oo are the roots of the equation 90 2 — 19A9 + D = 0, is 
1 (A® — BD 
D 1 co — co' 
it — 3co'v) 2 — oo [u — —, w^ 
+ 
A*/ - BD 
(t — 3twv) 2 — co (^u — — w'j 
then by similar reasoning it is concluded that 
D = +, 25A 2 — 9D = +, A= — , N = —, character is 5r, 
D = + , 25A 2 -9D = + , A = -, N= + ,1 
D = + , 25A 2 - 9D = +, A = +, l „ r + 4d. 
D = + , 25A 2 — 9D — —, j 
(Hermite’s second set of criteria.)