DDNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 363 



where the last transformation is valid since we are dealing with abso- 

 lutely convergent series. Also : 



1 ^~"* 1 



(58) x""'-'-^^ 2 x'^'-'-^-'" 



5=0 



(59) ^^.,/= '2 ^^^.'-- 



5=0 



Adding (57), (58), (59), and changing slightly the summation on the 

 right, we have : 



1 J. *=^ J=^i 



(60) -^.,,_: + -*^.,.+ 2 2 ^^''^- 

 ^ ^ i=l j=i 



1 r, «="' ri r, 



= - ^k. /-1, + - -A-, /, + ^ - -t, /-I, ?+l + - ^A-, /, ^,+1 



5=0 L 



i=TO j=e 



+ 22 «>..., ■ 



i=i .7=1 -I 



The series on the right is an absolutely and uniformly convergent 

 series of continuous functions. If we replace the terras by their values 

 given in (18) and (19), we can also write the series in the form : 



d ^^ d _ ^^ d 



(61) Tx"'-'-''^ ^, ^^^■•'•^+^~ ^, Ix^''"''' 



5=0 5=0 



which is the series of derivatives of the terms of (25). From this it 



follows that, if we differentiate the series for z^.^ term by term, we shall 



obtain an absolutely and uniformly convergent series of continuous 



functions ; and tiierefore z,. i has a continuous derivative at each point 



of our sub-interval, which is precisely (61), and this, as we have seen, 



is the same as the right side of (60). We can therefore write (60) in 



the form 



d 1 r^ *^'" •^-'' o 



di ""'■ ' ^ X ""'■ '-' + X ^'' '+22 *^-'' ^'•> 5 



t=l j-\ 



and now giving h and I all possible values we have precisely the system 

 of equations (16). We have, then, the following result: 



In any suh-intcrval of c, not including the point a; = 0, the functions 

 Zji,_i represented by the series (25) are continuous in x, have continuous 



