766 
MR. J. C. MALET ON A CLASS OF INVARIANTS. 
As another example let ns suppose y 1 and y 2 connected by any relation ; for con¬ 
venience take y 2 —/(log Vi) and let us seek to determine how the coefficients of the 
equation are related, and also its solution. 
Transforming as before we have 
1 + 2Q, +Q 3 =0 
/'«+2/m+QV(o=o 
from which 
on MS'® 
n A*)-fit) 
A)-fit) 
from this We have 
iv / ivfo={ v say 
hence 
«=F- i {|v?r^} 
and the complete solution of the equation is 
y= Ae F “{J Vfi& } +B/{F- {\v%dx }} 
to determine the condition between P 1 and P 2 we have 
f+ 4 Q ]Q3 t 
Q,* 
substituting F~ l {j\/ fVi.'ej for t in the left-hand side of this equation we get the 
required result. 
It is to be remarked that J\/P fix is an invariant* since if the substitution x=<p(t) 
removes the second term from the equation we have 
P s dx= 
where u has the same meaning as before; 
From the equation J=0 which expresses the condition y^y^= 1 we can derive the 
linear differential equation of the third order, of which the solution is 
y=ky-?+~By£+C yi yz 
where A, B, and C are arbitrary constants. 
