IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 57 



A2W, then, or, written in expanded form, 



^ d 2 u ^ du ( dan 1 dA\ 



* 2U ~ fj aii d^i + fi dxi\d^ ~ 2A aii dx } )> 



is an absolute covariant. We notice that the terms involving second 

 derivatives are identical in A 2 u and in L(u), so that the latter may be 

 written 



L(u) = A 2 w + 2 d% 



i 



du 

 dXi 



au, 



. s^ dan 1 ■v A dA ,.„. 



di = ai -^d^ + 2A4 a «dx-; (45) 



Similarly, the transformed differential expression, L(u), may be 

 written 



L(u) = A 2 u + > di r— + au. 



; d^i 



Now since, when L(u) goes over into L(u), A 2 u, au go over into A 2 u, au 



respectively, it follows that ^. di —- goes over into 2 di — , in other 



i 1 dxi c ~i dki 



words that it is an absolute covariant. 



Hence we conclude that the d's are transformed contragrediently to 



the — 's. The expression 2 A^djdxi, then, is of the form of (37), 

 page 43, and is, therefore, a relative covariant of weight two ; or 



2 W^ ( 46 ) 



i 



is an absolute covariant, if we define U by the formula 



k = j 2 Ai i d i } ^ 47 ) 



;' 



Since (46) is an absolute covariant, the Vs must be transformed con- 

 tragrediently to the dx's, 



h = 2 5? '*• (48) 



This being the case, the expression 



3 S -£)**»• (49 > 



