754 
therefore 
ME. J. C. MALET ON A CLASS OF INVAEIANTS. 
dQ 1 
dx 
or substituting the value of Q x we get 
■flMHr—.* 
the required condition. 
To solve the equation in this case, remembering that Q x =— we at once 
see that y 2 is the solution of the equation 
from which 
a/'(*)2 +(2PiA*)+/»}y=° 
e-fa** 
and the complete solution is 
2/3 y/ffa) 
p — \dx 
where A and B are arbitrary constants. 
*y 
If now in equation (4) we suppose that f(x) is not known, replacing f(x) by y we 
cioa 4*Ta n rt a! nti a-a a-P 4-V\ r\ A/viirrt4-i Arv 
.(5) 
2 
£)■-» 
IS 
A^+%2 
where A 
B, C, D are arbitrary constants and y lt y % the solutions of the linear equation 
Again it appears 
that if y=(f)(x) satisfies equation (5) the complete solution- of it is 
A<f>(x )+B 
V C <f>(x)+D 
Let us now consider the two equations 
1+2%2+p^o 
<fay . o -n d v 
dx 3 
d?y t ^.p, dy 
^ +2Ri i +R ^^° 
and 
