MR. J. 0. MALET 0N A CLASS OP INVARIANTS, 
763 
and therefore we have for the complete solution 
dx 
y 
If 
A0(*)+B>//(x)+C x (a;)+D } 
<j> (x) = e az , \jj(x) = e Px , x{ x ) =&yx > 
Qi, Q 3 , Q 3 are constants determined from the equations previously given; in fact we 
easily find 
4 Qi=— (a+/3+y), 6Q 3 =a/3+ay+/fy, 4Q 3 = — oc(3y 
H, G, and I are constants, and the solution is 
y=e 
<x+/3+y 
A o°-x 
{Ae“*+ BeP*+C&*+ D} 
II. 
In addition to the class of invariants of Linear Differential Equations which I have 
discussed in the first part of this paper, there is also another class worth noticing, 
namely, functions of the coefficients of the equation which remain unaltered when the 
independent variable is changed. I propose now to consider these functions. 
If we take the equation 
and let x=cj)(t), the new equation is 
where 
The quadratic , 
S+®,|+p*=» 
2Q 1 =2P 1 f-^, Q a =P 2 f* 
From these values of Q x and Q 3 we easily find the identity 
§ +4Ql Q 2 f + 4P lP> 
Hence we see that 
Q 2 
, (14) 
is an invariant of this kind of the equation of the second order. 
