DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 389 

 or, what is the same thing, 



XQ.-. , _ '-;- 



- r.o i.o r 



Now, let 



W * = 5? + 2!0-2! P ' rf**-' 



. .. . .- . , . . .' (4) 

 symbolically, then the coefficient of 



n r tl 

 in the foregoing expression is 



r 



r!I 3? 



the summation extending to those values of r and t that leave r + t the same 

 throughout, that is, the coefficient is 



r + t'. 

 and, therefore, 



J -'. _ * 



n-s! ,, n-0\e\ 



When * is changed into n * and the quantities C are introduced from (3), this 

 takes the form 



If it were desirable the summation on the right-hand side might be extended to the 

 value 6 = n, for C,^,,. vanishes if m < m', 



12. The only place where, as yet, the zero value -of P l has been admitted is in the 

 definition of W,, but no essential use of that value has been made ; now, however, it 

 will be found that the removal of the terms involving P t and Q l from (i.) and (ii.) 

 materially simplifies the analysis. Writing (iii.) in detail for the lowest values 

 of s and using the condition Q t = 0, we have in succession 



(5)-, 



(6)', 



and so on, the number of equations being . 



