320 MK. L. X. C. F1LON OX TIIK UKSISTAXCK To 



5. Alternative Solution for tin 1 xnn>e Section*. 



There exists also an alternative solution ; it is generally of a less convenient form 

 than the one last given. It may, however, be useful in certain cases. 

 Return now to the boundary conditions (4') and write 



, sinh 2? sin 2 (n e) 

 w iv t -f Ire" - 



cos 2y 



we then obtain 



(lit/* - _ _. . 



= when rj ^ p or f) := p , a < < + <* 



and 



<lic. ., f . cosh 2a . 



-^- + ^TC" < sin '2rj -\ sin '2 (rj e) > = 0, 



when 



The latter condition may be written 



-^ + ^TC~ sin 2 ( e) - : - -f cos 2 (77 t) . sin 2 = 0. 



''f cos *y 



Now let us write 



W| C^ CTj |- 73*2 



where 



i i wi n ^i (w^ 



" ' 'V " ' rff " ' * ? : 

 aud 



= 0, >j e = rb 7 dnt./dr) = 0, 77 = y. 

 But 



^CT! , . , x /cosh 2 + cos ^7 cos 2e\ 



~/fc + ITC- sin 2 (r? e) - - - ) = .... (1 2) 



''f \ cos 2y I 



'** + Arc- cos 2 (T; - e) sin 2 = . . . ... . . . . (13) 



A . , w7r . n + Tr(^-e 

 rs, = i, A,,suih - -sin 



,. o 2y 2y 



when 

 Assume 



i "i*-o -i ' (7r f nir(ij e) 

 2 B,, smh - - cos ***i 



1=1 7 7 



we have now to express between limits y 



/ ; 7T0 . 1V0 WTT , 



cos '20 = (' + ^j cos -f bi cos --+...+ b. cos - + . . . 

 77 7 





