405] 
AN EIGHTH MEMOIR ON QUANTICS. 
153 
The two new Tables are: 
Table No. 83. M=(*'lfx > y)-. See 143. 
Table No. 84. N = (*$&, y) 4 . See 143. 
Article No. 255.—Formula) fur the canonical form aa? + hf + cz 5 = 0, where x+ y + z = 0. 
255. The quintic (a, b, c, d, e, /$#, y) 5 may be expressed in the form 
ru 5 + sv 5 + tic 5 , 
where u, v, w are linear functions of (x, y) such that u + v + w = 0. Or, what is the 
same thing, the quintic may be represented in the canonical form 
ax 5 + by' + cz 5 , 
where x + y + z — 0 ; this is = (a — c, — c, — c, — c, — c, b — c^x, y) 5 , and the different 
covariants and invariants of the quintic may hence be expressed in terms of these 
coefficients (a, b, c). 
For the invariants we have 
G = J = b 2 c 2 4- c 2 a 2 4- a 2 b 2 — 2abc (a + b + c), 
Q = K = a-b' 2 c 2 (be + ca + ab), 
- U = L = a 4 6 4 c 4 , 
W = I — 4a 5 b s c 5 {b - c) (c — a) (a — b). 
[Observe that throughout the present Memoir, the invariants, instead of being called 
Gr, Q, — TJ, W are called I, J, K, L, viz. the I, J, K, L in all that follows denote 
the invariants, and not the covariants denoted by these letters in 142, 143. Moreover 
D is used to denote the invariant Q r , which is in fact the discriminant of the quintic.] 
Hence, writing for a moment 
a + b 4- c =j), and therefore J = q 1 — 4pr, 
be + ca + ab = q 
abc = r 
= r 
we have 
(a - b) 2 (b - c)°- (c - a) 2 = p 2 q 2 - 4çr - 4p s r + 18pqr - 27r 2 , 
and thence 
/2 = 16r 10 (p 2 q 2 — 4 q s — 4p 3 r +18pqr — 27 r 2 ), 
and 
J ( K 2 - JL) 2 + 8K 3 L - 12JKL 2 - 432Z ;i 
= r 10 1($ 2 — 4pr) 16p 2 + 8q s - (q 2 — 4pi') 72q — 432r 2 }, 
= 8r 10 {(q 2 — 4pr) (2p 2 — 9q) + q s — 54?’ 2 }, 
= 16?’ 10 \p 2 q 2 — 4 q 3 — 4p 3 r + 18pqr — 27 r 2 ], 
C. VI. 
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