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144] A THIRD MEMOIR UPON QU ANTICS. 313 
We have also the following relations between L, L', &c. and a, b, c, d, e, f viz. 
aP — bM + cN = 0, 
aM' + bP' -2cM+3dN = 0, 
aA r ' + 25Jf — cL’ . + SeN = 0, 
3 bN' . - dL' + 2eM+fN=0, 
3cN'-2dM'+ eP' + fM = 0, 
dN'~ eM'+/P = 0. 
The quartinvariant No. 19 [G] is equal to 
-AG + B\ 
i.e. it is in fact equal to — 4 into the discriminant of the quintic No. 14, [A]. 
The octinvariant No. 25 [Q] is expressible in terms of the coefficients of Nos. 14 
and 16, viz. A, B, C, as before, and ¿a, (3, 7, ¿3 the coefficients of No. 16, [D], i.e. 
a = 3 (ace — ad 2 — b 2 e +2bed — c 3 ), 
/3 = acf - ade — b-f + bd 2 + bee — c 2 d, 
7 = adf—ae 2 — bef + bde + c-e —cd 2 , 
3=3 (bdf— be 2 + 2ede — c 2 f — d 3 ), 
then No. 25 is equal to 
A, B, C 
a , ¡3, 7 
/3, 7> $ 
The value of the discriminant No. 26, [Q'], is 
(No. 19) 2 —128 No. 25. [that is Q'= G 2 —128Q.] 
We have also an expression for the discriminant in terms of L, L', &c., viz. three 
times the discriminant No. 26 is equal to 
[or say 3Q' =] LL' + 64MM' — 64A r A r/ , 
a remarkable formula, the discovery of which is due to Mr Salmon. 
It may be noticed, that in the particular case in which the quintic has two square 
factors, if we write 
(a, b, c, d, e, fjsc, y) 5 = 5 {(p, q, rjx, yf} 2 . yfycc, y), 
C. II. 
40