of equal order and argument 47 



2?t(l-a) /37r\^ ( nOL\\: ^ 



where m — ( — J , w 



IT \naj 



X 



we use Stokes' asymptotic formula* 



cos IItt (mw + 10^)] dw ~ 2 " ^ (3m) " * exp | - ir (|w)^}, 



.'0 " 



valid for large values of m. 

 This process gives 



exp { - in 2^ a ~ ^ (1 - g)^ | 



{2a(l-a)}V(27rn) 



^" ("«)'- r^ ., Mi ./ 



Hence ./nOi«) ^ /JgLVe^xM 



where 



X («)= V(l - «•-■) + log « - log {1 + V(l - «"^)l +i«"'(2 - 2a)i 

 Since % (i) = -02047, 



a rough approximation to .7io„o (500)/J.iooo (500) is (f )* e-"'*l 



5. We now prove the monotonic properties (valid for $ ^ ^ tt) 

 stated in §§ 2, 3 : 



(A) To prove that /^ (</>) = i (6<^)* sin ^ . (1 -cos ^)-- is a 

 decreasing function, we have 



d ... .,.,.., _ (3 + 2cos^)<^^^,(^) ' 



re ^^^' ^^^/^ ^- - — (i-cos^)3 — ' 



where g, (6) = [5 sin 0(1- cos ^)/(9 + G cos 9)]- 6 + sin 6, 



so that 



(y/ ((9) = - 6 (1 - cos ey/{9 + 6 cos 6)' ^ 0, and ^r (0) = 0. 



We now see that gi{0)^0, and so f/{<ji)^0, which is the 

 result stated. 



(B) To prove that 



/,{<!>) =^-^ [(8/6*) - {(^^ sin 6/(1 - cos Of]] 



is an increasing function, we first prove two subsidiary theorems, 

 namely : 



B (i). If c = cos 0, s^ sin 0, then the function 

 g, (0) = (85 + 163c + 84c- + 18c^) (/> - is (1 - c) (149 + 157c + 44c0 

 is not positive. 



* Math. Papers, ii. p. 343. The result may also easily be derived from 

 NicholBon's expression of Airy's integral in terms of Bessel functions of order ± J, 

 Phil. Mag., July 1909, pp. 6—17, 



