Mr Watson, Bessel functions of lar^ge order 107 



and so 



I i sinh t tan /3 + (cosh t — l)\^sm^^ sec /S V[2 (cosh it — cos v)} 



^ tan/9 I sinh ^t\. 



That is to say \ i sinh t tan /3 + (cosh i — 1 ) j exceeds botJt, j cosh ^ — 1 j 

 and also tan /3 | sinh ht\. 



In order to simplify the subsequent anafysis, it is convenient 

 to place a restriction on /S. We shall coiisequently assume in 

 futui^e that O^/S^Itt, so that tan^^^l. This restriction is not of 

 importance so far as the final result is concerned, because Debye's 

 formula, quoted in | 1 (ii), is effective whenever sec/3^1 + S, 

 where 8 is any positive constant ; and so it is certainly effective 

 when sec /3 ^ \/2. The importance of the analysis in the present 

 investigation is due to the fact that it is valid for small values 



of ;8. 



10. Consider what happens when r ^ ^, whether v, V are both 

 positive or both negative. 



When I 2'| ^f , we have (on the T-contour) 



T = I iiTHanyS + irT' \ ^\T'\{^ + ^\T\)< h, 

 and if I i I < I, we have (on the ^-contour) 

 T = I [i tan /3 (cosh ^ — 1) + (sinh t — t)]\ 



00 



$ t |«|'"//?i!^e*-l -1 = 2-12 -1-75 <^. 



Hence, when t ^ |-, we must have both | ^j ^ f and \t\ ^ f . 

 But, when | T | ^ | , we have 



I (dr/dT) \ = \T\.\itan/3+iT\^\T\.\^R(T)\^^\T\"-^^\. 

 Also (as in § 4) when | ^ | ^ f , we have 

 j (dr/dt) \ = \i sinh t tan /3 4- (cosh t— 1)\ 



^ I cosh ^ — 1 I = cosh u — cos v^2 sin- (fjr ^2) = 0-137, 



and so j (dt/dr) | ^ 7-3. 



From these results we see that, when r^^, 



\ d (t -T)/dT\< 15 <57r. 



We shall make use of this inequality in § 12. 



11. Consider next what happens when ^t ^^, whether v, V 

 are both positive or both negative. 



When \T\^2, we have (on the IT-contour) 



Also, when \t\ ^2 and v + /3 ^ ^ir, we have u ^ ^tt \/3, and 

 then 



T = u — sec /3 sinh u cos (v + /3)^u^ 1. 



H_-2 



