66 



Proceedings of the Royal Irish Academy. 



Hence 



Lt 



P [u) cos Ou dti \ dO = 7r 



J 



and, by similar reasoning 



Lt 



r 



Jo 



P (u) sin 0*i du 



P (v) 



d6. 



dv, 



This result holds good even though P (v) becomes infinite at certain 

 points, provided that [P(4')] s is integrable from to oo ; which is the same as 

 saying that K{t) may become infinite, provided that [K (t)J is integrable 

 between and 1. When this is not so, we must examine the general 

 ease r > 1. 



We have 



1r 



g 



=f If 



t«K(t)dt 



da 



Lt r + «! 



£ = », 



j:;iu> 



cos 6u du 



n p(« 



-Jo 



) sin Ou du 



d.0. 



The general term of this expression is 



Lt Jj_ 

 j=« q\ si 



+ f r 



roo ~i iq r~ poo ~ |2S 



P(u) cos Oudu P(u) sin dudu\ dO 



Jo J L J J 



where j + s = r ; this, omitting a constant numerical factor, is the same as 



Lt 



" " " ?(Ki) . . ■ ^(^37) P(lhqn) ■ ■ ■ P(thr) COS 0U X ... COS 0W 2? 



sin 8ih qil ... sin 0u, r du x . . . du lr 



dd. 



The product of an even number of cosines or sines may be resolved into 

 a sum of cosines ; hence this integral breaks up inLo others of the form 



Lt 



P{ ",)... P(« m ) 



cos (?*,... + Kj, - Mp« - ... - w m ) ^Mi . . . du„ 

 which is the same, omitting a factor 2, as the following : — 



dl 



r°° 



Lt |... 

 E=»Jo . 



P(w,) . . . P(u m ) 



sinf(w, + 



+ m„ - M, 



</> — «7>+l 



Wm) 



f=ooj J «j + . . . + Op - W p+ , - 



Now, putting 



«! = V lf M] + ttj = Vi, . . . Uri + . . . + Up = 



dux . . . du m . 



Vi = r i>+i> u p*i "•" w p+= = V" 



Upn + . . . + u m = v m , 



