544 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
313. The expressions for the coefficients (a, b, c, d, e, f) are in the first instance 
obtained in the forms 
a= 2(L +5MC+10C 2 ), 
b=-2(I/ + 3M'C)A 
c= 2(L +MC-2C 2 )A-', 
d= — 2(1/— M/C), 
e= 2(L-3MC + 2C 2 )A- 2 , 
f = — 2(L' — 5M'C)A -1 , 
where, developing M. Heemite’s expressions, 
72L= 
24M= 
24L'= 
24M'= 
A 7 B + 1 
A 4 B - 1 
ABI +1 
1 + 1 
A 6 C 2 + 1 
A 3 C - 1 
Cl +5 
A 5 B" + 6 
A 2 B 2 - 3 
A 4 BC - 24 
ABC +12 
A 3 B 2 + 9 
A 3 C 2 - 39 
C 2 +24 
A 2 B 2 C + 9 
ABC 2 +108 
C 3 + 72 
and substituting these values, we find 
36a = 
36 b = 
36c = 
36d= 
36e 
= 
36 f= 
A 7 B + 
1 
A 2 BI - 3 
A 6 B + 
1 
ABI - 3 
A 5 B 
+ 1 
BI -3 
A 6 C 2 + 
1 
ACI -24 
A 5 C + 
1 
Cl -12 
A 4 C 
+ 1 
A 5 B 2 + 
6 
A 4 B 2 + 
6 
A 3 B 2 
+ 6 
A 4 BC - 
39 
A 3 BC - 
27 
A 2 BC 
-15 
A 3 B 3 + 
9 
A 2 L 3 + 
9 
AB 3 
+ 9 
A 3 C 2 — 
54 
A 2 C 2 - 
42 
AC 2 
-30 
A 2 BC - 
36 
too 
B 2 C 
+ 36 
ABC 2 + 
288 
BC 2 +144 
C 3 +1152 
! 
1 have not thought it worth while to make in these formulae the substitutions A=J, 
B = — K, C=9L+JK, which would give the expressions for (a, b, c, d, e, f) in terms of 
J, K, L. 
314. Substituting for (x, y) their values in terms of (X, Y), we have 
(a, b, c, d, e,fjx, yf 
=(«• << *’/X 2 4c (^+Q^) x + 5?e( w-Mk) t. 
{w~ v /X ) x + r/c(-|s+ p ^ 5 ) Y )’ 
=(X, jM., v , v\ yJ, 7/)(X, Y) 5 suppose, 
and by what precedes 
ax' 1 -j- f3xy -f- yy 1 — v 7 AXY ; 
