102 Proceedings of the Rojjal Irish Academy. 



that no value of {p+ v\-)a-/v is very small; hence, if la is large enough, 

 all the values of 



l(2^ + i>\')ayv',Xi5Iayr>y-, or (jj + rX'-//(rP/3'-) 



can he as large as we please, and hence 



2(7/3/ vX'-^p .Vii 



so large that the approximate forms of the I functions for large values of the 

 parameter may he applied as accurately as we please, and it thus appears 

 evident that, under such circumstances, all the values of 7; are given 

 approximately hy (15). 



Aet. 18. ABigorous Proof of last Proposition. Nv/niher of Roots in a Circular 

 Contour of large Piadius having Origin as Centre. 



A rigorous proof of the last statement presents some difficulties, however. 

 Let p) be any quantity, in general complex, not restricted to a value 

 which satisfies the period-equation, and denote 'p + vA^ hy p)' ; then, if la is 

 sufficiently small 



2i//3/-/ .Y|4 2(//3/-/ Vf*..., ,, ,. ,.^. , 



in the sense that the cliference between the left- and the right-hand members 

 can be made less than any assigned quantity by taking la small enough ; for 

 the difference may be made less than a certain multiple of ^Ui-ji^vp')^ as 

 follows from the binomial theorem. If, under these circumstances, with the 

 origin as centre, there is described a circle for which 



mod 2« (-//v)i = {r + J-jTT, (28) 



r being zero, or any integer, it may be proved that the number of roots of the 

 period-equation within this contour is r. (The ckcle might equally well be 

 taken so that the right-hand member of (28) is any other quantity lying 

 between r-rr and (/' + l)7r, and finitely different from both.) Let the equation 

 be written in the form 



uHmi-^i:^^^ - i\(:y.)] \M{v.^ + ^[v^)\ - f^iO^^) -ii(^'.)} [i^M^O 



-f/i(«Oj] = 0. (29) 



A comparison with (8) shows that in this form the proper equation has been, 

 for convenience, multiplied by uM'^-^- 



With a view to examine the increase of argument of the left-hand member 

 as p)' describes the circumference of the circle, we first trace the changes in 



