IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 31 



L(u) by (f>. For if L(u) go over under u = -r\ into a canonical form 

 A(t/), <fiL(ii) will go over under the same transformation into (f)A(r)), 

 which is likewise canonical. We shall be inclined, then, to expect that 

 the conditions in question will consist in the vanishing of expressions 

 which are "invariants of the differential equation." And such proves 

 to be the case. 



We examine the question first for an expression of the second order. 

 Let 



L(u) = 2< a vj^:.+ Z a i— t" aw 



go over, by u = ■ v, into 



^* dxidxj 



dxidxj 

 d 2 V 



dx; 



t. j 



tV^j 



+ 2 



dr] 



If this is to be canonical, we must, by (19), page 18, have 



2 2% 3% * 



to 



If 



A = 



+ ai = 0, 



an 



«i» 



1,2, 



^0, 



to. 



a>m\ 



a 



mm 



1 1 fi 



these equations may be solved for A > i = 1, 2, ... to. Note 



OX{ 



here that A is an invariant of the differential equation. The solution 

 in question will be 



dlogfl 

 to 



2 A 



(24) 



let us say. Necessary and sufficient conditions that these equations 

 possess a solution log are 



di<j 

 dxj 



%- = 0, 



dxi 



i, j =1,2,. 



TO. 



(25) 



The expression on the left of this last equation is an absolute inva- 

 riant of the differential equation. For, first, K it k^ themselves are abso- 

 lute invariants for a multiplication of L(u) by <j). Next, if L(u) go 



