MR. J. 0. MALET ON A CLASS OF INVARIANTS. 
755 
and let us seek the condition that the two solutions of the first equation, y x and y 2 , 
should be connected with the two solutions of the second, y 3 and by the relation , 
Let f(x) be the common value of these fractions, then referring to equation (4) we 
have at once the required condition, viz.: 
diq 
dx 
It may be remarked that if H does not equal 0 but a constant, say k, the complete 
solution of the equation 
3 + «p,* + p*-o 
is 
y=e~ J PldaJ {A sin \/kx -\-B cos y/kx} 
The cubic. 
Let us now consider the equation of the third order 
g+^g+ar.l+P*-* 
we have two invariants G and H; and the equation becomes by removing the second 
term 
g+SHg+G,-. 
Let us now consider what relation must exist between H and G in order that two 
solutions of the equation y 1 and y. 2 should be connected by the relation yi—y % x. 
Transform the cubic by substituting yy 2 for y and let the resulting equation be 
S+3Q 1 g+3Q,|+Q 32/ =0 
Since this equation is satisfied by y= 1 and also by y=x we have at once 
Q 3 =0 and Q 2 =0 
dQ 1 
hence 
-Qi 2 
2Qi 
dx 
=H 
3 
from which we find 
dx 3 
=G 
G-f+ 2 HQ 1= 0 
