409] ON THE CONDITIONS FOR THE EXISTENCE &C. 305 
12. The conditions for the common [cubic] factor are 
a, 
46, 
6c, 
4 d, e 
= 0, 
a, 46, 
6c, 
4 d, 
e 
b, 
4c, 
6d, 
4e, / 
6, 4c, 
6d, 
4e, 
/ 
the several determinants whereof are given in Table No. 27 of my “Third Memoir on 
Qualities,” Philosophical Transactions, vol. cxlvi. (1856), pp. 627—647, [144]. 
13. These determinants must therefore vanish, for (A, B, G) = 0, and also for 
(21, 23, ... £, S JOÎ) = 0, that is, they must be syzygetically connected with (A, B, G), and 
also with (21, 23, ... 8, 93Ï). The relation to (A, B, G) is in fact given in the Table 
appended to Table No. 27, viz. this is 
Cx + _Z? x 4-Ax 
1234 = 
+ 6 a 2 
— 12 ab 
+ 16 ac — 10 6 2 
1235 = 
+ 6 ab 
- 2 ac - 10 6 2 
+ 6 ad 
1236 - 
- 2 ac + 8 6 2 
+ 6 ad — 18 be 
- 2 df+ 8 e 2 
1245 = 
+ 18 ac 
— 6 ad — 30 6c 
+ 8 ae + 10 bd 
1246 = 
+ 12 6c 
+ 4 ae — 4 bd — 24 c 2 
+ 4 be + 8 cd 
1345 = 
+ 24 ad 
— 8 ae — 40 bd 
+ 4 af + 20 be 
1256 - 
— 1 ae + 4 bd + 3 c 2 
+ 1 af + 5 be — 18 cd 
— 1 bf + 4ce+ 3d 2 
2345 = 
+ 20 ae + 40 bd — 30 c 2 
— 80 be + 20 cd 
+ 20 bf+ 40 ce - 30 d 2 
1346 = 
+ 4 ae + 8 bd + 6 c 2 
— 36 cd 
+ 4 6/+ 8 ce + 6 rf 2 
2346 = 
+ 4 af + 20 be 
— 8 bf - 4 ce 
+ 24 cf 
1356 = 
+ 4 be + 8 cd 
+ 4 bf - 4 ce — 24 d 2 
+ 12 de 
2356 = 
+ 8 bf + 10 ce 
— 6 cf — 30 de 
+ 18 df 
1456 = 
+ 6 ce 
+ 6 cf — 18 de 
- 2 df+ 8 e 2 
2456 = 
+ 6 cf 
- 2 df — 10 e 2 
+ 6 ef 
3456 = 
+ 16 df- 10 e 2 
-12 ef 
+ 6/ 2 
14. Between the expressions 21, 23, &c., and 1234, 1235, &c., there exist relations 
the form of which is indicated by the following Table: 
t 
C. VI. 
39