ME. J. 0. MALET ON A CLASS OF INVARIANTS. 
753 
Let us now seek the condition that the two solutions of the equation y—y\ and 
y—y 2 should be connected by the relation y^ytfc, which relation, since it depends 
only on the ratios of y 1 and y 2 , must be expressible in terms of the invariant H. 
If in the equation we change y to yy 2 the solutions of the resulting equation will be 
y= 1 and y—x ; hence if the equation is 
g + *4f + Q*-o 
we have Q 2 =0, Q x =0 remembering now that 
Q,-Q 1 2 -f=H . 
the required condition is 
H=0 
To solve the equation in this case we have 
Hence y^=e~\ Pldx and the complete solution is 
y=e~\ Vldx { Ax +B } 
where A and B are arbitrary constants. 
We may remark also, as is at once obvious, that the condition H=0 is the eliminant 
of the equations 
dx 3 
+aP 1 *+F«M> 
and 
dx 
+Pj2/=0 
If we seek the mere general condition that the solutions should be connected by 
the relation yi=y 2 f(x), where/(as) is some given function of x, transforming as before 
by the substitution of yy 2 for y the resulting equation 
g+^f+Q^o 
must have for solutions y— 1 and y=f(x ), hence we have 
Q 2 —0 and/ /, ( £C )+ 2 Qi/ / ( £C ) = ° 
5 d 2 
