630 
MR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 
16, viz. A, B, C, as before, and ^a, j3, 7 , the coefficients of No. 16, i. e. 
a, = ^{ace—ad^— h^e -{- 21 ) 601 — d) 
(3= acf — ade — hd^ -\-bce — cV 
7 = adf — ae ^ — bcf-{- bde-{-de — cd‘^ 
h = 3(bdf—be^ -{-2cde— &f —cP), 
then No. 25 is equal to 
A, B, C 
a, [3, 7 
(3, 7, ^ 
The value of the discriminant No. 26 is 
(No. 19)^—128 No. 25. 
We have also an expression for the discriminant in terms of L, L', &c., viz. three times 
the discriminant No. 26 is equal to 
LL'+64MM'-64NN', 
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 
then 
(a, b, c, d, e,f\x,yy=b{{p, q, r\x,yyY{\, (dyix,y), 
a=5'Kp^, b=4pqX-{-p^()j, c={2q^-\-pr)X-{-2pq(jtj, 
e—r^\-\-4qr^, d=2qr'k-\-{2q‘^-{-pr)iJj •, 
and these values give 
P=K(6^^— jor) P'=K( lOy^— 15jor) 
M = K. 10 /?^ M'=K. 10 ^/’ 
N = K . bp^ N' = K . br^, 
where the value of K is 
— 2qiJj7-^r'K^y{pr — q^y . 
The table No. 29 is the invariant of the twelfth degree of the quintic, given in its 
simplest form, i.e. in a form not containing any power higher than the fourth of the 
leading coefficient a ; this invariant was first calculated by M. Faa de Bruno. 
