IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 49 



Lagrange's Identity, and these two expressions are, therefore, in accord- 

 ance with the proposition noted on page 16, mutually adjoint. 

 It remains, then, to prove that 



1 d& BS 



1 u ^^ 



i,1 



1 J dxi dii 



rj dtj \J dxi J 



or, what is the same thing, that 



2^.( Jd ^J -o- 



1 1> . 



Now the coefficient of Si in this equation vanishes. For -= — 1S 



J OXi 



equal to i#, the cofactor in 



dii 



' OL 



of 



dxj 

 9& 



So that the coefficient in question, viz., 2a7V ir^ )» * s 



■ acy\«/ OXi J 



equal to ^ ^» an ^ this expression vanishes. For ■—- is the sum 



j Of / Olj 



of the m— 1 determinants obtained by substituting in t# for the 

 elements of each of its columns in turn the derivatives, with regard to 

 £j, of the elements of that column. Consider any one of these m— 1 

 determinants, 



