232 
m. A. CAYLEY ON THE POEISM OE 
or representing this by 1+2B^+2C|^+2D|’+ &c., we have 
Cf+Df +Er+Fr+Gr+Hf +If +J|9+K|'“+L|”+Mr^+ • • 
+(d^^-cr) 
'■l-2 6S-2ac 
^'+ 8 abc 
+ 4 a'c' 
— 24 a'6c' 
CO 
J 
$'+ 64a'6c' 
$^+ l6a‘‘c'* 
+ 4 6' 
-8 6' 
— 24 a6'c 
+ 64 a6'c 
+ 96 a'6'c' 
-320a'6'c' 
—320 a'6'c' 
+ 16 6" 
— 32 6' 
— 160 ab*c 
+ 384 a6'c 
+ 960 a W 
1 
+ 64 6' 
— 128 6^ 
— 896 a6'c 
l60a<6c^f. 
+ 256 6® 
+ 1280 0^6 V i 
— 2688 a'M I 
+ 2048 aVc 1 i 
— 512 6* 
+ a'‘( — d'i^+^c'd^ — c^s®) ri— 6 6^ — 6 ac 
1 +24 6' 
{ 
1 
|' + 48 abc 
^'+ 24 a'c' 
^"-240 a'6c' 
80 a'c' 
-80 6' 
— 240 a6'c 
+ 960 a6'c 
+ 1440a'6'c' 
+ 240 6" 
-672 6' 
— 3360a6"c 
+ 1792 6' 
^'’+&cr 
+ 0^2 - 6 c-d'%'^ + 6 cV^'* - 2 / 1 - 1 0 6^ - 1 0 ac 
+ 60 6 ' 
^'+120a6c 
—280 6' 
"•} 
+ a''(-5c?T- + &c.){l+&c.} + &c. 
And the values of the coefficients C, D, E, &c. thus are 
C 
D 
E 
F 
G 
H 
I 
J 
K 
L 
a~'c 
+ d 
-2 bd 
— 2 acd 
— 1 a'rf' 
+ 6 a'6rf' 
+ 6 a'ce?' 
+ 2 a"rf' 
- 20 a"6r7' 
— 20 a'cc?' 
- 5 
— 1 c' 
+ 4 b^d 
+ 8 abed 
+ 6 a'c'rf 
— 5 a'c" 
- 48 a'6c«^' 
- 30 a"c'<7' , 
+ 120 a"6'J' 
+ 240 
+ 2 6c' 
+ 2 ac' 
— 24 ab^cd 
-24 aW 
— 20 a'c'«? 
+ 240 a'6'crf' 
+ 300 a"6c'c? 
+ 140 
-8 6'rf 
— 8 a6c' 
— 36 (i‘bc^d 
+ 80 a'6'rf' 
+ 160 a'6c'(7 
+ 70 a"c"c? 
— 560 
— 4 6'c' 
+ 16 b*d 
+ 64 ab^cd 
+ 144 a'6'c'<7 
+ 14 a'c' 
-960 a'6'cd' 
-1800 
+ 8 6'c' 
+ 24 a6'c' 
+ 30 a'6c" 
-240 a'6"6?' 
- 800 a'6'c'^^ 
— 700 
-32 b^d 
— 160 a6"cc? 
-480 a'6'c'«? 
— 112 a'6c' 
- 42 
-16 6"c' 
— 64 a6'c' 
-120 a'6'c" 
+ 672 a'6'<7' 
+ 3360 
+ 64 6't? 
+ 384 ab‘’cd 
+ 1440 a'6"^^^ 
+ 3200 
+ 32 6'c' 
+ 160 a6"c' 
+ 400 a'6'c" 
+ 560 
-128 6='rf 
- 896 a6'c<; 
-1792 
— 64 6'c' 
— 384 a6'c' 
-4032 
+ 256 bH 
-1200 
+ 128 6V 
+ 2048 
+ 896 
~ 256 
+ 1 
-3 
+ 4 
-3 
+ 4 
-19 
+ 60 
-124 
+ 214 
-4 
But in the sequel only the coefficients up to I are made use of: the expressions for J, 
K, L, M may however be useful, and they are given accordingly. 
The sums of the numerical coefficients are given here and elsewhere, as they are very 
useful for verifications; thus, putting a = h = c=d=l, we have, as should be. 
-v/l + 4|+8r + 12f=l + 2| + 2r(l, +1, -3, +4, -3, +4, -19, +60, -124, +214, -455,&c.Xl, 
Proceeding now to form the several terms of the matrix 
( C, D, E, F, G. . . ), 
I D, E, F, G, H, . . I 
which may be represented by 12, 13, &c., viz. 
12 = 
C, D 
13= 
C, E 
, &c.. 
