il99 



ƒ (.f) = A, II, (.0 ^A,H, (..) -f A^ 11^ (.r) + . . . 



Supposing this expansion to be possible tlie coefficients An may be 

 found by means of tlie relations (6) and (7) 



00 



An — -— — 1 ^ ^V'(«) Hn («) da. 



2" . n ! yjrj 



With these values the second member reduces to 



S = Urn 2 ^" An Hn i'V) ') 

 d—i 



where 



1 r* , ^^ &>' H„ Lv) H„ (a) 









Hence 



* (I— 02)/3'-! 



S = Lmt I f? •* (7/i I ƒ (^t) cos {<( — dx)3dii 



Ü 



or 





Now the second member of this equation i-eprcseuts /'(/'), wheu 

 this function satisfies the conditions of Dihichlkt betweeu the limifs 

 — GO and -}- GO. Every function of this kind may therefore be 

 expanded in a series of the functions H. 



8. We now proceed to give some examples of this expansion. 

 I. Let f{.r)^xi', then we Jiave 



xp = A,H, + A^ H^ + A, /ƒ, + . . . 

 where 



00 



-^ f.v"Hn 



An — —z^ I >>•>' Hn e-^"- dx . 



2" .71.' 



Evidentlv this integral is zero when .v/' //„ is an uneven function 



') The idea of inlrüdiicing ^ was suggested to me by Prof. P. Deb ye. 

 Proceedings Royal Acad. Amsterdam. Vol. XVI. 



