DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 367 



Now consider the following limit: 



(78) limit x~''< (log a;)~^'«~^^ y".^ ^ 



= limit 2 '^ ^i,k,iX~^''Q-Ogx)~^'<~^^zl'^. 



On the right, if we take the limit of each term in the summation, we 

 shall find that it is zero in every case except the one in which k ^ k and 

 / = e« (cf (64) III.). We have, then : 



(79) limit x-'< (logaO'^'-^'V--' = ^i,<,e^ ^^^ _^y/ 



So we can write as a set of solutions of (1), corresponding to the ele- 

 mentary divisors (r — r^Y'^ : 



(80) 2./ ::=.«(log.O^ ^/ (^;,^1,2,...J' 



where the functions i/'*' are continuous in the interval < a: < c, and 

 such that : 



(81) ' •A'^'^l n = ^ ^—^,Ai,e- 



\ J Vt 1^=0 (e^ — A) ! ' ' " 



The constants A^-^j are determined independently of the functions 

 a, y in (1) ; and therefore all that we have said on page 346 in regard 

 to certain sets of them as linearly independent solutions of the equations 

 (4) in tlie special case of a,y = holds equally well here. 



In order to obtain solutions corresponding to (r — ?'k)^*,* we assumed 

 in the treatment of the canonical system that \h\j\ jlog xf^~ was inte- 

 grate up to aj = 0. Now, since the coefficients Vj^^^i are linear functions 

 with constant coefficients of the coefficients ajj, it will be sufficient, in 

 order to obtain solutions of (1) corresponding to (?• — r^)'", to assume 

 that |a,y| I log x^'^~ is integrable up to ic = for all values of i andy. 



Our results may be stated as follows : If (r — ^k)^" i& cm elementary 

 divisor of il(^r), and if we consider (dl the elementary divisors (r — r^Y* of 

 A(r) such that Rr^^=. Rr^, and denote by e^ that exponent ichich is as 

 great as any other exponent in this set of elementary divisors, and assume 

 that I a,j\ I log X f^~ is integrable up to x = 0; we can develop e^ solutions 

 ofO): 



* Cf. p. 349. 



