168 Miss Dorothy Wrinch on an Asymptotic 



y-HM-i 



N 2 = f * e±^«+*+*>(&* 



v/ — 7T 



y + ^-H 1 



N 9 



,= f 77 g±«^r+«+*) € ±»i(«+i8+i)^ & t^. 



2-" 2 



t/ — 7T 



r+/3+ 



x^^-^""^".- 8 *^-^/^^"^ 



dO. 



We are still at liberty to choose S as we please, provided 

 only that pjB and 8 tend to infinity with p, the factors 

 e ±™(.Y+ +sf) being adopted according as (9 is positive or 



negative, and the factors g— 7r ^° ^ *' being adopted ac- 

 cording as <p — 6 is positive or negative. 



We shall therefore, if possible, choose 8 with a view to 

 ensuring the existence of critical points for the integrals 

 iN lf iN 2 , 2 N r 2 N 2 . 



Critical values of 6 will exist if there exist roots of the two 

 equations 



7 r- iO / \ A -0—0-1 



»[ s '« T ±(?)«n 



We therefore choose 8 = p 2 , which gives the only possible 

 case. 



Consider first the integrals 1 N J and 2 N 2 . Critical values 

 are given by the equation 



= |+2/*7T. 



There will be one, and only one, for each integral, and it 

 will be 



'-I- 



Hence, using Stokes's result as before, 



AT -*C«+/3+y+a+i) -«|(o+j8+y+^+l) 



i^li — ^ e 4 



i(p 



"4~ / * 2 \ 



J— | 



/S- -J(a+/5 + r + c+S/2) -i|(a+^-i- 7+c - + s/2) 4pV** 

 = V lit . p e * e 



