MR. J. 0. MALET ON A CLASS OF INVARIANTS. 
757 
If we have the two conditions G=0 and H=0, then 
V\=y^=y^ 
and the complete solution is 
y—e~\ Pld *{ A+Ba;+Che 2 } 
G=0 expresses the condition that y=e~\ Pldx should be a solution of the equation. 
If we wish to find the conditions necessary that the following relations should exist 
between the solutions of the cubic 
Vr=yM x \ y*=yM x ) 
where and \fj(x) are known functions of x ; change as before the cubic by sub¬ 
stituting yy% for y and let it become 
we must evidently have Q 3 =0, hence we have 
we have also 
Now if we let 
and 
f"+3Q 1 f'+3Q 8 f=0 
f // +3Q 1 f / +3Q^ / =0 
q 2 -q> 3 -^= h 
Qi 8 -3QiQ a -gj=G 
3 
we have Q 1 =F(ic), Q.,=f(x), and the conditions sought are 
H+F'(a)+(F(*)) 3 -/( a; )=0 . 
G+F» + 3F (x)f(x )—(F (x)) 3 —0 
and 
(7) 
( 8 ) 
To solve the equation in this case we have 
