A\) 



In this section vve will give their principal results though not 

 altogether after their methods, and make some additional remarks. 



13. 3Ir. Galbrun considers the question of the expansion of a 

 function between the limits a and h, in a series 



where 



1 1 r 



A, = -—-—[ e~-\f{a) Th («) da. 

 2"n/ \/„J 



He finds that this expansion is possible when f {a-) satisfies the 

 conditions of Dirichlet between the limits a and h. This agrees 

 with our result in Art. 7, the only difference being that our limits 

 were — oo and -\- oo. This difference however is not essential, for 

 considering a function which has the value zero for all values 

 a ]> X ^ h Art. 7 gives immediately the expansion of Mi*. Gat-brun. 



His proof rests on two interesting relations which may be easily 

 deduced from the formulae in the first part of this paper. 



Tiie first relation 



^ Hj, {x) H,, {a) _ 1 //„+! (.r) //„ {a) - //„ [x] JI,j^, {a) 



(29) 



may be established in this way. 

 Accoi'ding to (5) we have 



2.r//„ (.;) = Iln+x (.^0 + 2^/^-1 (.'•) 

 2aHn («) = lIn+\ («) + 2«//;, - 1 («) 

 Multiplying these equations by //„(«) and H,,{a^ wc find by sub 

 tract ing 



{n > 0) 



2 (.r - «) Tin {x) //, («) r= H,J^, {.r) /ƒ„ («) - Iln {x) ƒƒ„_ 1 («) 



- 2n [II, (x) H,-x («) - //,._i {x) Ih («)]. 

 Hence, putting for n successively 0, 'J,2,..7i, we get 



2.1/ 

 1 



F727 



2« . nl 



2(.r-«)F^(.r)if,(«)=H,(.r)£f,(«) -^.(.r)/^.,(«) - 4[F (.r)^,(«)-^,(^-)^.(«)] 



2(.^•-«)F,(.^•)^„(«)=://.+l(.r)//,(«)-^„(•^•)^"+>(")- 



-2n[H,(.r)i^„ _ , («)-i7"-H^)H,(«)J . 



