312 
A THIRD MEMOIR UPON QUANTICS. 
[144 
No. 28. 
N. 
M. 
L. 
L\ 
P. 
P'. 
M'. 
N\ 
a?df + 3 
a 2 ef + 1 
a 2 / 2 + 1 
a 2 / 2 + 3 
a 2 / 2 ... 
a 2 / 2 - 1 
abf 2 + 1 
acf 2 + 3 
a 2 « 2 - 2 
abdf+ 2 
abef — 34 
abef — 22 
abef + 1 
abef + 9 
acef + 2 
adef— 9 
abcf — 9 
a6e 2 — 9 
aedf + 76 
aedf— 12 
aedf — 3 
aedf— 1 
ad 2 f— 9 
ad + 6 
abde + 1 
cic 2 f — 9 
ace 2 — 32 
ace 2 + 64 
ace 2 + 2 
ace 2 — 18 
ade 2 + 6 
b 2 f 2 - 2 
ac 2 e + 18 
ac<ie + 32 
ad 2 e — 12 
ad 2 e — 36 
ad 2 e 
acPe + 12 
b 2 ef - 9 
beef + 1 
acd 2 — 12 
ad? — 18 
b 2 df- 32 
b 2 df+ 64 
b 2 df + 2 
b 2 df- 18 
bedf + 32 
bd 2 f+ 18 
b 3 f + 6 
b 2 cf + 6 
b 2 e 2 + 225 
b 2 e 2 - 45 
6 2 e 2 — 9 
b 2 e 2 
bee 2 
bde 2 - 15 
b 2 ce — 15 
b 2 de ... 
bc 2 f- 12 
bc 2 f - 36 
bc 2 f ... 
bc 2 f + 12 
bd 2 e — 15 
c 2 df — 12 
b 2 d 2 + 10 
bc 2 e - 15 
bede— 820 
bede + 20 
bede + 31 
bede +45 
<?f - 18 
c 2 e 2 + 10 
bchl ... 
bed 2 +10 
bd 3 + 480 
bd 3 
bd 3 - 18 
bd 3 - 30 
c 2 de + 10 
cd 2 e 
c 4 
c 3 d 
c 3 e + 480 
c 3 e 
c 3 e - 18 
<?e — 30 
cd 3 
d 4 
c 2 d 2 - 320 
c 2 d 2 
c 2 d 2 + 12 
c 2 d 2 + 20 
If the coefficients of the table 14 are represented by \A, B, \G, viz. writing 
A = 2 (ae — 4bd + 3c 2 ), 
B = of— 3 be + 2 cd, 
G = 2 (bf- 4ce + 3d 2 ), 
then we have the following relations between 1234, &c. and A, B, G, viz. 
C x 
+ B x 
+ Â X 
1234 = 
+ 6 a 2 
— 12 ab 
+ 16 ac — 10 b 2 
1235 = 
+ 6 ab 
- 2 ac -10 b 2 
+ 6 ad 
1236 = 
— 2 ac + 8 b 2 
+ 6 ad — 18 be 
- 2 df + 8 e 2 
1245 = 
+ 18 ac 
— 6 ad — 30 be 
+ 8 ae + 10 bd 
1246 = 
+ 12 be 
+ 4 ae — 4 bd — 24 c 2 
+ 4 be + 8 cd 
1345 = 
+ 24: etch 
— 8 ae — 40 bd 
+ 4 af + 20 be 
1256 = 
— 1 ae + 4 bd + 3 c 2 
+ 1 af + 5 be — 18 cd 
- 1 bf + 4 ce + 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 bf + 8 ce + Od 2 
2346 = 
+ 4 af + 20 be 
— 8 bf — 4 ce 
+ 24 cf 
1356 = 
+ 4 be + 8 cd 
+ 'l 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 
and the following relations between L, L', &c. and A, B, G, viz. 
Gx 
+ B x 
+ A x 
N = 
— 3 ac + 3 b 2 
+ 3 ad — 3 be 
— 1 ae + 1 bd 
M = 
— 3 ad + 3 be 
+ 3 ae — 3 c 2 
— 1 af + 1 cd 
L = 
+ 11 ae + 28 bd — 39 c 2 
+ 1 af - 75 be + 74 cd 
+ 11 bf + 28 ce - 39 d 2 
IJ = 
— 7 ae + 4: bd + 3 c 2 
+ 3 a/+ 15 be- 18 cd 
- 7 6/+ 4 ce + 3d 2 
2 P = 
— 1 ae — 2 èc? + 3 c 2 
+ 3 be — 3 cd 
+ 1 bf + 2 ce — 3d 2 
F = 
+ 3 ae — 6 bd + 3 c 2 
— 1 af + 1 cd 
+ 3 bf — 6ce+ 3d 2 
M'= 
— 1 af + 1 cd 
+ 3 bf - 3d 2 
- 3 cf + 3 de 
N' = 
— 1 bf + 1 ce 
+ 3 cf — 3 de 
- 3 df+ 3 e 2