PEOFESSOE CAYLEY’S EIGHTH MEMOIE ON QUANTICS. 519 
Article No. 255. Formulae for the canonical form ax 5 -\-by 3 -\-cz 5 = 0, where x-\-y-\-z=0. 
255. The quintic ( a , b , c, d, e, ffx, y) & may be expressed in the form 
ru! 1 -f- sv 5 + iw\ 
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 b -\-cz 5 , 
where x-\-y + 2 = 0 ; this is =(a— c, —c, —c, —c, —c, b—cfx^yf, and the different 
covariants and invariants of the quintic may hence be expressed in terms of these coeffi- 
cients ( a , b , c ). 
For the invariants we have 
No. 19 =J =5V+ c 2 a 2 -\-a 2 b 2 — 2abc{a-\-b-\-c). 
No. 25 ='K=a 2 b 2 c 2 {bc-4-ca-\-ab). 
-No. 29 =L =a 4 b 4 c\ 
No. 29A=I =4a 5 b 5 c\b-c){c-a){a-b). 
Hence, writing for a moment 
a -\-b -\-c =p, and .-. J =q 2 —4pr, 
bc-\-ca-\-ab=q K =r 2 q, 
abc —r L =r\ 
we have 
(a — b) 2 (b — c) 2 (c— a) 2 —gf<f — 4q 3 — 4]fr + 1 8 pgr —27 r 2 , 
and thence 
I 2 = 1 6r ,0 ( p 2 q 2 — 4 q 3 — 4 p 3 r + 1 8 pqr —27 r 2 ), 
and 
J(K 2 - JL) 2 + 8K 3 L— 72 JKL 2 - 432L 3 
= r 10 { (q 2 — 4pr)l 6p 2 + 8^ 3 — (<f — 4 pr) 72q — 432r 2 } 
= 8r 10 { (q 2 — 4pr){2p 2 — 9q) + q z — 5 4r 2 } 
= 16r ]0 {q) 2 q 2 —4q 3 —4q) 3 r+18j)qr—27r 2 }, 
that is, 
I 2 =J(K 2 — JL) 2 +8K 3 L— 72JKL 2 — 432L 3 , 
which is the simplest mode of obtaining the expression for the square of the 18-thic or 
skew invariant I in terms of the invariants J, K, L of the degrees 4, 8, 12 respectively. 
No. 26=D={b 2 c 2 +c 2 a 2 +a 2 b 2 -2abc(a+b+c)} 2 -128a 2 b 2 c 2 (bc+ca+ab), 
=q 4 —8q 2 jyr—12 8 qr 2 + 1 6+V, 
D=Norm((J<?) i -F(m)*+(«^) i ). 
And we have also the following covariants : 
No. 14=(— ac, ab—ac—bc , —bcfx, y) 2 
= bcyz + cazx + abxy. 
