DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 397 

 and, therefore, 



Jr(\ 

 ll W 



~ " ' ~ ~ ., m-l!r-m 



It will be noticed that the coefficient of the first term is 1 (r + *) e/i 1 . The 

 quantity d'Q t /dz r , which has a dimension number (r + *), will be multiplied by z' 1 "*"' 

 in the invariantive equation, i.e., by 1 + (r + s) e/* 1 ; and the first term in the product 

 of the two will thus be d'YJdyf. The correctness of this result furnishes a slight 

 indirect verification of (13). 



As particular examples of (13), corresponding to those of (12), we have 



* (I - V) - 3 e/ xP 3 - 8f (,?,) i (n + 1) 





