31B MR. L. N. U. FILOX OX THK 15KSISTAXCK TO 



4. Analytical Work for Sections Bounded by One Elliptic and Two 



Hyperbolic Arcs. 



Consider now the transformation 



x = c cosh f sin rj, 

 y = c sinh cos 17. 



If we allow rj to vary l)etween ft and ft 1 , and between + and a, the point 

 (x, y) will move within the space contained between the ellipse 



I ' _, i 



c* cosh s <~ sinh- x ~ 

 and the two hyperbolas 



*? if- jf ^ 



Using then the coordinates (f, TJ) instead ol (x, y) we find that our equations for w 

 Income 



- + l v = for { > (:3'). 



, / 1 2 rf- ^+ f ^ \. i 



I ^^ /^* ^^ %~ ^* f-f I 



v ^^ o ^^* x 



Also 



-rr + ^rc 2 sin 2>/ =: 0, when f=i, /8'<^<^| 



-J-Tr 2 sinh 2f = 0, when -n = Bor B', a < < a 



f'?7 J 



Write now 



, sinh 2f sin 

 ir = ;, ire" 



cosh 2 

 Then 



(5). 



Let us assume 



'TV A i ^+l/ , 2n + \ir(n -e)\ 2n+ 



i'i = 2 ( A., sinh (r) e) + B,, cosh - -' sin - 



=o >. 2a j 2 



where e = | 



Then conditions (5) and (3') are identically satisfied. Let us now determine the 

 coefficients A and B so that (6) shall also be satisfied. 



