34 BELL SYSTEM TECHNICAL JOURNAL 



it follows from a formula exactly analagous to (3.3) that 



so that the problem is reduced to the solution of the definite integral 

 G n (x, y). 



In the integral (3.6), write g = l-f/z, so that 



/, = Vl+P 2 sin 2 M -l, (3.8) 



whence 



G n (x,y) = sin x — / cos ftx • cos 2«/t • cos(y sin ^)^m 



"^ (3.9) 



2 /" r/2 

 + cos x • — / sin hx • cos 2n\i • cos(y sin \x)d\i 

 ■kJq 



= P„sin x-\rQ n cosx, (3.10) 



where P n and Q n denote the definite integrals of (3.9). This effects 

 a further reduction of the problem to the solution of the definite 

 integrals P n and Q n . 



In the integrands of these integrals expand cos hx and sin hx in the 

 usual power series, and in each term thereof introduce the expansion 



h s = f^-J (sinV) (l+a,ip 2 sinV+a.2P 4 sin V+ • • • ), 



where the coefficients are given by 



a-=(-iy, (5+2i ~ 1) -7 1 V 



By aid of this procedure it is easily shown that 



D _ , . (pV2) 2 d* (. 2 d 2 , A d* \ . 



+ 



{p\x/2Y d* (, „ ,d*„ A d* 



d s /\ , & , d d* \t t \ 



4! dy*\ K dy 



( P V2) 6 #» /- . . ^ 4 ^ 



r 2 (l-^P 2 ^+a 62 p 4 ^- ..)/■.(?) 



6! dy 2 \ D1K df 

 + . . , (3.H) 



with a corresponding expansion formula for Q n . 



