and Random Flights in one, two, or three Dimensions. 339 

 for it may be written 



2r r°° . 7 /sinr t i'\ 2 I °° sin rx 7 



I smsxdl l = — l -cossxax 



7tsJ \ rx J 7tJ x 



1 r x sin( s + r)x— sin (s — r)x 7 i A 

 = — 1 i J ax = 1 or 0, 



7Tj ( X 



according as s is less or greater than r. 



In like manner for a second lemma, corresponding with 

 (25), we may reason again from the triangle GFE (fig. 1). 

 J (V) is replaced by smg/g, a potential function symmetrical 

 in three dimensions about E and satisfying everywhere 

 V 2 + 1 = 0. It may be expanded about Gr in Legendre's 

 series * 



. sin<? . /sine cos e\ , 



A,- r +A^- ? 7-)+- 



fi being written for cos G, and accordingly 



C 



dfjb 



sin \/(e 2 +■'/*— 2ef p) _ . sin e 



S(*+f*-*2efiL) 



When E and F are interchanged, the same integral is 

 seen to be proportional to sin///, and may therefore be 

 equated to 



, sin e sin f 



A» 



•$:■ 



where A ' is now an absolute constant, whose value is deter- 

 mined to be unity by putting e, or f, equal to zero. We 

 may therefore write 



sin V (e 2 -t f 2 — 2efii) _ sin e sin /' {Ki) , 



As in the case of tw T o dimensions, similar reasoning shows 

 that 



T +1 cos \Z(e 2 + f 2 — 2effju) sing cos/ (KVCS 



^^ *(*+f-itfk —r-T' ' ( } 



provided «</. 



With appropriate changes, we may now follow Kluyver's 

 argument for two dimensions. The same diagram (fig. 2) 

 will serve, only the successive triangles are no longer 

 limited to lie in one plane. Instead of the angles 0, we have 

 now to deal with their cosines, of which all values are to be 



* < Theory of Sound,' § 330. 

 2 B 2 



