752 
MR. J. C. MALET ON A CLASS OP INVARIANTS. 
Now H, G, I and the remaining coefficients of equation (2) are in a certain sense 
invariants of the original equation (1), for they remain unaltered if in equation (l) we 
change y to yf(x), where f(x) is any function of x, and then divide by f(x) so as to 
make the coefficient of unity; thus writing the equation so transformed 
d n y i d n ~^y , n,n—l.~ d n ~*y , ^ 
df d*f 
and for convenience writing f{x), &c .,f f\f" we find 
O _/±?i/ n _ / /, + 2P 1 /+P a / 
V 5 l- y J Vs~ y 
(3) 
Q 3 — 
Hence also 
from which we easily prove 
and 
/ / " + 3P L T+3P a /+P»/ 
/ 
dQ 1= ff^ff dP, 
dx f* ' dx 
d?Qi ■ 2/3 ^ Pi 
dx* f f ^ dx* 
dx 
- -Ql— op 3 —3P P -UP 
' 1 1 3 & 
-6Q i 4 +12Q 1 3 Q 2 -4Q 1 Q 3 -3Q/+Q 1 -^ i =I 
^Qi S SQ^g+Qg ^ 
In a similar manner we find 
The theorem proved in these cases may be easily shown to be generally true, as 
follows :—If from equation (3) we remove the second term by the substitution of 
ye~ I Qld * for y it is evident the result must be identical with (2); but the coefficients 
of the resulting equation are the same functions of Q 1? Q 2 , Q 3 , &c\, as the coefficients 
of (2) are of P 1? P 2 , P 3 , &c., hence the theorem is proved. 
I proceed now to some particular applications of the general theorem. 
The quadratic. 
dfy 
da? 
+2P ‘!+p^=o 
d£ x 
Here we have the invariant H or 
