40 Mr. J. H. C. Searle on the Effect of 



Put, therefore, A — = n. 



Hence, on substituting the values (16) in (B) and neglecting 

 terms of order A 4 , we have 



2 + 2 cos n( 1 + ~- Jcosh n (l— ~7 ) — ^ 2 sin n sinh w = 0. 



/. 2 + 2 cos n cosh n — 2n j- (cos n sinh rc + sin rc cosh n) 



4 — A 2 sin n sinh w = 0. 



Now take 



n = n + r), ... .... • (I 7 ) 



where n satisfies 



cos n cosh n + 1=0, . .... (18) 



and 7] is small. 

 .*. 2 + 2 (cos rc — ?7 sin n Q ) (cosh w + 17 sinh n ) 

 — 2rj (sin rc cosh w — cos n sinh n ) — 2w 7 ( cos ^0 sinh r <0 



+ sin n cosh w ) —X 2 sin n sinh /i = 0. 

 .\ using (18), 



2 . 



w \2 ( cos rc sinh w + sin n cosh n H — sin n Q sinh rc |^ 

 4 ( sm 7? cosh n — cos y/ sinh n j 



I 



77 \2 & tan w -f tanh n + — tan n tanh w 

 (_ tan n — tanh n Q 



By (18) 

 cosh n Q — ~sec7? . 

 ..-. sinh ?? = +^an t? , according as n Q lies in the 3rd or 2nd quadrai 

 ,\ tanh n = ± sin w , „ ,, „ 2nd or 3rd „ 



Thus, 

 according as 5 is even or odd where w approximates to 



