MR. J. 0. MALET ON A CLASS OF INVARIANTS. 
765 
to solve the equation in this case, we have 
dx* 
from which 
and the complete solution is 
Q2 = I ) 2^ /3 or 
t=eW^ F * d * 
y —Acf -J- B 
where A and B are arbitrary constants. 
Again, let us seek the condition for fay*—I, transforming as before so that y Y —e\ 
we find now that the coefficients of the transformed equation are connected by the 
relations 
1 +2Q 1 +Q 3 =0 
4 — 4Q 1 +Q2 ==: 0 
therefore 
2Q 1= 1, Qa= — 2 
and we have 
J 3 +2=0 
or 
(f+ 4P >P S )+2P/=0 
for the required condition. 
In this case we find as before the solution to be 
More generally if y x =y^ we have, using the same transformation as before, the 
following equations connecting the coefficients of the transformed equation 
1 + 2Q-L +Q 2 =0 
m 3 +2mQ 1 +Q 2 =0 
from which 
2Qi=—(m+1), Q 2 -m 
and we find as before for the required condition 
the solution being 
y = Ae~^ V? * dx +He 
dP 
— 3 +4P 1 P2) — 4(m-f-l) 2 P 2 3 ^0 
this investigation fails, as it should, when m= 1. 
