180 
AN EIGHTH MEMOIR ON QU ANTICS. 
[405 
where, developing M. Hermite’s expressions, 
72 L = 
24 J/= 
2477 - 
24JT = 
A 7 B 
+ 1 
A 4 B - 1 
dÆ/ + 
1 
1+ 1 
A*C I 2 
+ 1 
A 3 C - 1 
Cl + 
5 
A 6 JP 
+ 6 
A 2 B 2 - 3 
A 4 BC 
- 24 
d,SO + 12 
A 3 B 2 
+ 9 
C 2 + 24 
A 3 C 2 
- 39 
A 2 B 2 C 
+ 9 
ABC 2 
+ 108 
C 3 
+ 72 
and substituting these values, we find 
36a 
36b = 
36c = 
36d = 
36e 
= 
36f= 
ylVi + 
1 
A*BI - 3 
A*B + 
1 
ABI - 
3 
A°B 
+ 1 
BI - 3 
A 6 C 2 + 
1 
AGI - 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 B 3 + 
9 
AB 3 
+ 9 
A 3 C 2 - 
54 
A 2 C 2 - 
42 
AC 2 
- 30 
A 2 BC - 
36 
BC 2 + 
144 
B 2 C 
+ 36 
ABC 2 + 
288 
C 3 + 
1152 
I 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, 
(a, b, c, d, e, f^x, y) 5 
i Q' 
= (a, b, c, d, e, /J —^ + 
1 i-F 
X + 
w-,A z + 
2 VO 
1 
and by what precedes 
this gives 
and thence 
2VOVJ. * ‘ 2VO 
= (X, fi, v, v, \x!> X'\X, Yf suppose, 
ax 2 + ßxy + 7y* = "J A X Y ; 
ddy- — ßdyd x + yd x 2 = — V Ad x d y , 
Y), we have 
h~ qU ) 7 ‘ 
■f Ä + l^Ä)Y 
(ad,/ — ßdyd x + yd x 2 y (a, b, c, d, e, f y)° 
= Ad/dy (X, ya, v, y!, v, X'\X, Y) 5 
— 120A (vX + v Y) ;