596 TRANSACTIONS OF SECTION A. 



If also 1 + n.,x- + ^l^.^■^ + . . . = (1 + Xo.t- + A^.r' + . . .)" ^ 

 then p = | + = (cos«-|-asma)-^(;x,-5 + g- . . .) 



- ^^ tan + ^^ (tan a - - tan' o) - ^ (tan a - - tan^ a + -; tan'' a) . . . 

 ^3 n> 6 JV 3 5 



wliere the orders decrease similarly. 



Case 2 : n greater than z. 



If 71 -z cosh ^, ji not being small, 



J_„ (s)-cos «7r J„ (s) = sin wtt a/ - o~' (n not integral) 



Y„(s)= _,y'2jlt--' (» integral) 



Then -I" siuh /3 = 1 - -^ eoth"^ -h -' coth' /3 . . . 



t = n ^tanb /S - f:i) - J log,2 - i (u, - ? ^3 + \ 1^1 . . .) 



+ 1 (;i^ coth ^ - ^^^. coth' /3 + ...) + -' J: --l|-;+-^^; • • . 



where the B's are the Bernoullian numbers, and orders decrease as before. 



Case 3 : when n and s are nearly equal, and fairly large, J„(s) may be reduced 

 to an Airy's integral ' in the form 



j^_ (,)_l(6y £%os {..' + .-=@%r},Z,. 



and its tabulation may be conducted at once by the use of tables given for this 

 integral by Airy and later by Stokes.^ 

 In particular 



J„(«) = rQ)2~3 3-;r-'« J 



This is correct to three places of decimals when n - 8, and to four when n = \Q 

 so that it is not necessary for 7i to be very large. 

 Similarly 



J.„(/O = r0 • 2' 3-i TT-' «-' cos ('"r-^) 



More generally, if 



Y„(")=-rQ)2-3^«-J 



F(p) = cos (w' + pw) dw 

 Jo 



f{p) = 1 sin («■' + piv) dw 

 Jo 



' Airy, Caml>. Phil. Trans., vi. p. 379 ; vlii. p. 595. 

 - Moth, and Phys. P/i2>ers, ii. p. 329 et se^. 



