758 MR. J. C. MALET ON A CLASS OE INVARIANTS, 
thence y z must be the solution of the equation 
J+{Ei-F(«)}y=0 
_ e~l p i dx 
y= -Wf 
and therefore the complete solution of the cubic is 
from which we have 
or 
y= 
e -\Pidx 
(*V-*"*7 
-SA+Tty{x)+Ctfx)} 
3 
If we make and then eliminate xjj(x) between equations (7) and (8) we 
should evidently obtain the condition necessary that the relation y^—y§ should exist. 
I have, however, obtained the condition in a much more simple manner, and found it 
to be 
2G- 
which result I give further on in the present paper. 
Let us now consider the more general problem; to determine the relation between 
the coefficients of the cubic in order that two solutions should be connected by the 
relation y l =y 2 f(x) where f(x) is a given function of x. 
As before, change y to y 2 y, and the equation must become of the form 
^ +8Qi ^ +3Q *i-° 
Q 3 vanishing. 
We have then 
/ /// +3Q 1 / ,/ +3Q 2 / / =0 .... 
. (9) 
Also 
. 
.... (10) 
2Q/—3Q,Q 3 -fJ=G. 
.... (11) 
and the problem is to eliminate Q x and Q a from these equations. 
If we substitute in (10) and (11) the value of Q 2 found from (9), we find they can be 
written in the forms 
