336 



.MR. L. N. G. FILON ON THE RESISTANCE TO 



- -r * (^ + i 2 sin 2/8 ) = re 2 sinh 2 tan 2/3 - 8r (cosh 2a + cos 2/3) 

 J f yrff / 



/ 2 - 1 > IT 8 S 



X 



(2n 



cosh 



2/8 sinh 2 



cosh 2f + cos 



- 16/3 8 ] cosh -- 



Hence the equation giving the maxima and minima is 

 tan 2 sinh 2 



(cosh 2 + COB 2/8) (cosh 2 + 2 cos 2/3) M = (2 + !)TT 

 cosh 2f + cos 2/8 =0 (2n + 1)- TT* - 



. , 



2/9 



cosh 



2n + 



2 (cosh 2 + cos 2/8) = 

 cosh 2f + cos 2/3 , 1=0 



sinh [2rt + ITT -I- 4/3] ^ sinh [2n + ITT - 4/9] -|- I 



2/9 P! i 2?H- 



(2n + l)7r-4/3 



sech 



2/3 



One root of this equation is =0. Now it is clear that since S" is zero, and 

 therefore a minimum at = a, if it be not a maximum at f = 0, then there must be 

 a maximum somewhere between = and = a. 



To find whether = is a maximum or not, we have to investigate the sign of 



i.e., we have to investigate the sign ol 



1 



T 



sm 



cPw 1 dJ 



n0 \~] 



C " 8in 2 ^) J 



all the other terms vanishing when = 0. 



Now ( -rr + ^rc 2 sin 2/8 ) being always positive, we have to investigate the 

 J \rtf / 



sign of 



* t 



w en * = 0> 



1 dJ /dw 



sm 



and therefore ultimately the sign of 



