370 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 corresponding number — — -: , and the resulting solutions may be 



written : 



(90) yi^ x^< (log xr<-' E-' (x 1 1; 2; ; ; : I) ' 



where ^^'^ |_,^o == r« (r« — 1) . . . (r^ — n + 1 + i), 

 E''^\ _„= 1. 



By means of (83) we can now return to solutions of the equf,tion (82) 

 with the following result: 



If r^ is a root of the characteristic equnticn^ and \pi\ | log ccj^^~ is 

 integrable up to x = 0, where e^- ^ f^ for all k's such that JBr^ = Rr^, 

 e^ being the multiplicity of the root r^, then the eqiuition (82) has e^ 

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

 derivatives : 



(91) / -.A ^ ^ (A= 1, 2, . . . e«) 



dx' V » ^ n-i 



where the functions E^^_^ are continuous in the neighborhood of x =^ 

 and^ 



^n-i \x=(i= r^ (r^ — 1) . . . (r^ — i + 1). 



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 tliat: 

 b b 



I Gog ^') \Pi I ^^) I (log oc) \2h I dx 

 



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

 require that 



/ (log ^y I Pi I <^^, I (log xy \p2\dx 







converge.* 



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



