MR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 
629 
If the coefficients of the table 14 are represented by ^A, B, \C, viz. 
A = 2(ae — 46c?+3c^) 
B= af—dbe-\-2cd 
C =2(4^“4ce 
then we have the following- relations between 1234, &c. and A, B, C, viz. 
Cx 
+ BX 
+Ax 
1234 = 
+ 6 
— 12 ab 
+ l6 ac —10 6® 
1235 = 
+ 6 
O 
7 
1 
+ 6 ac? 
1236 = 
— 2 «c+ 8 
+ 6 ac? — 18 be 
- 2 df -V 8 e® 
1245 = 
+ 18 ac 
— 6 ac? — 30 be 
+ 8 ae +10 6c? 
1246 = 
+ 12 be 
+ 4 ae — 4 bd—24 c® 
+ 4 6e + 8 cc? 
1345 = 
+ 24 ad 
— 8 ae —40 6c? 
+ 4 a/ +20 6e 
1256 = 
— 1 ae + 4 bd-\- 3 
+ 1 of + 5 6e — 18 c<? 
— ] 6y + 4 ce + 3 c?® 
2345 = 
+ 20 «e+40 bd — 30 c" 
— 80 be +20 cd 
+ 20 bf +40 ce-30 c?® 
1346 = 
+ 4 ae + 8 bd+ 6 c" 
— 36 cd 
+ 4 6y + 8 ce+ 6 c?® 
2346 = 
+ 4 q/- +20 be 
— 8 bf — 4 ce 
+ 24 cf 
1356 = 
+ 4 be -j- 8 cd 
+ 4 bf — 4 ce — 24 c?® 
+ 12 e?e 
2356 = 
+ 8 + 10 ce 
— 6 c/ — 30 c?e 
+ 18 df 
1456 = 
+ 6 ce 
+ 6 cf — 18 de 
— 2 df -{■ 8 e® 
2456 = 
+ 6 c/ 
- 2 c?/ -10 e® 
+ 6 e/ 
3456 = 
o 
1 
+ 
-12 ef 
+ 6/® 
the following 
relations between L, 
L', &c. and A, B, C, 
viz. 
Cx 
+ BX 
+Ax 
N = 
— Z ac+ ^ b‘ 
+ 3 ad — 3 be 
— 1 ae + 1 6<? 
M = 
— 3 ad+ Zbc 
+ 3 ae — 3 c® 
— 1 af+ 1 cc? 
L = 
+ 11 ae+28 bd — 39 c“ 
+ 1 af — T5 6e+74 cd 
+ 11 6/ + 28 ce— 39 d~ 
L' = 
— 7 + 4 bd4r 3 c® 
+ 3 <y+15 be — 18 cd 
- 7 6/+ 4 ce+ 3 <?® 
2P = 
— 1 ae — 2 bd-\- 3 c® 
+ 3 6e — Z cd 
+ 1 6/+ 2 ce— 3 e?® 
P' = 
+ 3 ae— 6 bd+ 3 c® 
— 1 af \ cd 
+ 3 6/ — 6 ce + 3 c?® 
M' = 
— \ of \ cd 
+ 3 bf - 3 d~ 
— Z cf -{■ Z de 
N' = 
— \ bf + 1 ce 
+ Z cf — Z de 
- 3 df+ 3 e® 
We have also the following relations between L, L', &c. and a, h, c, d. e,f, viz. 
tfP — 6M + cN =0 
aM'+feP' — 2cM+3c?N =0 
«N'“}-26M'— cL' . +3eN =0 
H-36N' . - ^/L'+2eM4-/N=0 
+3cN'- 2<;M'+ eF +/M = 0 
+ d^'- eM'+/P =0. 
The quartinvariant No. 19 is equal to 
-AC^-B^ 
i. e. it is in fact equal to —4 into the discriminant of the quintic No. 14. 
The octinvariant No. 25 is expressible in terms of the coefficients of Nos. 14 and 
4 o 2 
