DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 475 



and all higher derivatives are zero to the order of small quantities retained. We now 

 have 



f * i / 1 \ 7 / t\ t 7\ 



Now, by 114, it follows that 



d* __ r ^ m / iV--' mlm l\ 



dsf* rmi r!r l!m rl 



for the relation (56), where, in the present case, 



- _ ad be 



~~ (d + a) 1 ~ r ** ' 

 and 



Hence, to the order of small quantities retained, it is necessary to consider on the 

 right-hand side of the transforming formula only the terms arising from r == m, and 

 r = m 1 ; and thus 



~ + lm(m-l) . { 

 Applying these equivalent operators to (57), we have 



....... (58). 



Similarly, if v f be the associate variable of rank p 1 and index ^p (n > p) 

 and if t] p be the same transformed associate, we have 



(59). 

 Again, if 6,. be an invariant of index p., and if 4> M be its transformed value, so that 



3 P 2 



