3Ir Watson, Besael functions of large order 109 



and observe that 



and also, in view of the fact that, as t varies through positive 

 vahies, t + T traces out in the Argand diagram a curve, through 

 the origin, whose slope obviously never exceeds V3, the distance 

 of all points of this curve from — 4^ tan $ must exceed 2 tan /3. 

 Hence | i tan /9 + ^ (T + ^) | ^ ^ tan /3. 



Using these two inequalities, combined with the fact that 

 j(T— i) j ^ 120|^|-/119, and the obvious inequalities 



\T+t\^\t\, I cosh t-l-^t'\^\t 1 V23, 

 |sinhi-^-i^-|^|^i-Yll9, 



we deduce from the last equation for T— ^ that 



\T-t\^l\t\ {120 1 1 \ll\^Y + 4 1 ^ IV23 + 16 U IV119 < U I'- 



Using now the inequality \T —t\i^\t\^ in place of 



I T- ^1^120 1^17119, 

 we get 



\T-t\^l\t\' + ^\t IV23 + 16 I ^ IV119 



< (1/24 + 4/23 + 16/119) \tY ^ ^\t\K 



Using now the inequality \T —t\^\\t\^,yNQ get, in place of the 

 last result, 



\T-t\% (1/192 + 4/23 + 16/119) | ^ 1=^ ^ ^ | ^ p'. 



From this result it follows that, when l^j^^, \T —t\% ^\t\, 

 and so iri^l^l^l. 



Consequently, from the formula for d{t - T)ldT given at the 

 beginning of § 9, we see that, when | ^ | $ |^, 



\dt^_dJ 

 dr dr 



i\t\ 



{i|«|.|(cosh«-l)| 



+ 



i (ii I ^ l)N 1* ^i^h ^ ^^^ /^ + (cosh t — 1)| 

 Now, when | ^ | < 2, 



1 po«?h /_l|>i|^|2ri_ 4 _ 16 _ 1>1|/|2 



and \smh.t\^\t\[l-l;-^- ...]>^\t\; 



and so, using the results of § 9, we get 



\d(t- T)ldr \ ^ 16/11 + (576/121) [4 (1/6 + 1/23) + 6/5] 

 <12, 



when 1^1 ■S^. . 



