764 
MR. J. C. MALET 0N A CLASS OF INVARIANTS. 
To arrive at this invariant directly, let us suppose that the second term is removed 
from the equation by the change of the independent variable ; we have then, supposing 
the transformed equation to be 
w +uy ~° 
Hence 
therefore 
2P 1 f-^'=0, w=P 2 <P 
|=^f +2W=fS (f + 4P 1 P i 
from which it is evident that 
^-^2 I ip p 
P* 1 ~i? dt 
^+4P P 
remains the same when the independent variable is changed. 
I shall denote this invariant by the letter J; and I now propose to give some 
examples of its use. 
Let us seek to determine what relation among the coefficients of the equation 
expresses the condition that the two solutions of equation (14) y x and y 2 should be 
connected by the equation y x y 2 = 1. 
Transform the independent variable so that e f shall be a solution of the new equation, 
then e~ f must also be a solution. Let the new equation be 
*+<'+«*=0 
and we have 
l + 2Q 1 -f-Q 3 =0 
1 —2Q 1 +Q b =0 
from which we find <^=0, Q 3 = — 1, but 
+4Q x Q 2 
Q* f 
= 0 
hence the required condition is J=0 or 
