248 Dr. J. W. Nicholson on the Asymptotic 



where 



f\{p) — 1 cos (iv +pw)thoi 



fo(p)=\ sm(w*+pw)dw, 



Jo 



f 3 (p)=r dive-^ + ^ (80) 



Jo 



Now Stokes * has shown that when a is not small, an 

 asymptotic value of 



/i(-«0-*/ s (-«r)= 1 ^e- iw3+ - w , 

 Jo 



is given by 



3±<ri e 



so that 



(81) 



/ '(- <r >=£i cos ( 2 (0°- i7r )' 



where o- is negative. 



Moreover, when p is positive, and not too small, 



(1)1" C ° S I (•*+'***"= oS? e "^ • ( 82 > 



The values of J n (z) and Y„ (z) o£ expansion C, when these 

 formulae are valid, become, on reduction, identical with 



«-,. (g)w . 



¥„(*), (^)*oob/) (83) 



* Camb. Phil. Trans, ix., Math, and Phj*. Papers, ii. p. 329 etseq. 



