106 Mr Watson, Bessel functions of large order 



In like manner, we find that 



dV _ (tan;e + F)^(tan/8 + ^7)^ 



^dU~ tan'/S + Ftan/S + iV- ' 



and it may be proved by quite simple algebra that the square of 

 this last fraction does not exceed 3. 



From the results just proved it follows on integration that 



\v\^\u\^/S, |F|^|i7!V3, 

 and hence 



h'!^iUI> \v\^^\t\^3, \U\^^\T\, |Fi$i|TlV3. 



9. We now return to the integrals of § 7. As in the corre- 

 sponding work of §§ 2 — 3, we have to obtain an upper bound for 

 \d{T — t)ldr i ; we shall in fact shew that this function does not 

 exceed 127r. 



We notice that formulae corresponding to those given in | 3 

 are 



{T-t){hAT+t)i\>?.u^-^^{T^^Tt + 1?)] 



= i tan yQ (cosh t-l-hP) + sinh t-t- ^t\ 

 d{t- T)/dT 



= {iT tan /3 + ^T']-^ - {{ sinh nan /3 + (cosh t - 1 )]-^ 



t-T 



" T [i sinh t tan ^ + (cosh ^ - 1)} 



^t{t-T) + i (sinh t - t) tan ^ +_(cosliJ^- 1 - ^i') 

 "^ T {% tan y8 + ^ T) (TsmhTtan yS + (coshl -1)} 



Now 



I i sinh t tan /3 + cosh t — 1\ 



= sec 13 V[(cosh u — cos v) {cosh v — cos (2/3 + w)|], 

 and since 



(cosh u — cos (2/3 + v)] — cos- B (cosh ii — cos v) 



= sin^ /3 (cosh u + cos v) + sin 2/3 sin v 

 ^ (1 + cos v) {sin- /S + sin 2/3 tan -^ vj 

 ^ (1 + cos w) {sin- 13 - sin 2y3 tan ^^] 



we have 



. I i sinh t tan /3 + (cosh ^ — 1) | ^ cosh w — cos « = | cosh t — l\. 

 Also 



cosh w - cos (2/3 + ?;) ^ 2 sin= (^8 + -iwi ^ 2 sin- 1/3, 



