370 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where p is a constant not zero. We may divide each solution by the 



corresponding number P , N , , and the resulting solutions may be 



(e K — X) ! 



written : 



(00) ,,*=»■ (log*)- E i ^ = 1,2,... J' 



where ^' A |^ = r K (r, - 1) . . . (r K - n + 1 + i), 

 E K ' X \ ft =l. 



By means of (83) we can now return to solutions of the equation (82) 

 with the following result: 



If r K is a root of the characteristic equation, and \p £ \ | log xj K is 

 iategrable up to x = 0. where e K ~> e k for all k's such that Rr k = Rr K , 

 e k being the multiplicity of the root r k , then the equation (82) has e K 

 linearly independent solutions which may be written with their first n — 1 

 derivatives : 



/'* = ;/* (log xf*~ K E*; K 



(91) J y «>* . . . ■ (X=l, 2,. ..e K ) 



dx i ■ v r n ~ l 



where the functions E*?_. are continuous in the neighborhood of x = 

 and, 



(92) • ^ l*=o 



•^n'-i \x=o— r K (r K — l)...(r K — i+l). 



Even for the equation of the second order this theorem does not 

 merely give the results of Professor Bocher's paper above quoted, but 

 goes a step farther, since in the case in which the two roots of the 

 characteristic equation are equal, we require merely that: 

 b b 



I 0°g x ) \Pi I dx > I ( lo g x ) \Pa I dx 

 *0 



converge, while Professor Bocher's method made it necessary for him to 



require that 



b b 



J (log xf I p x I dx, j (log x)~ \p 2 \dx 







converge.* 



* L. c. p. 48. The function xq x in this paper is the same as our p.^ 



