IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 17 



II. Change of Dependent Variable; Invariants and Cova- 

 riants. Invariants of a Differential Equation. 



§ 4. General Properties of Invariants and Covariants. 



We take up next the subject of the transformation of a differential 

 expression by change of dependent variable and of the invariants and 

 covariants of such a transformation. 



Taking our differential expression in the form (12), let it go over 

 under change of variable, u = ifr (x, y(- 77, into a differential ex- 

 pression A (77), with coefficients a. A (77) will be of the nth order, 

 and its coefficients, the a's, may be readily calculated. 



Formulas for the coefficients of the transformed differential expression, 

 p + q = n: a pq = a pq \}/. 



( W a«A 



p + q = n-l: a pq = w^a p+ i, q — + a Pt q+ i —\ + a p< rf. 



p -f q = n — 2: 



n(n - 1) / av av av\ 



° P9 - 2! V ap+2 ' 9 dx~ 2 + 2ap+1 ' q+1 dx^y~ + ° p,5+2 df) 



( a^ aiA 



+ (n — 1) I a p+hq — + a Piq+ i— J + a pq \{/. 

 p + q = n — k: 



4y (n - Q 1 a*-ty 



^ ~ jfg a ( n - *) ! * ! (* - l ~ ! aP+i ' 9+fc ~^ aa:%fc"^ ' > 

 For ordinary differential expressions these reduce to 



_ 4 (n - Q 1 „ d*-V 

 Gn " ft ~ £g (n - ife) ! (Jfe - ! <««*-» * U j 



while for expressions of the second order, ^ 



a 2 ^ ^ 



dxidxj ^ l dxi 



^ dhi ^ du 



I,] 



we should get 



vol. xliv. — 2 



