TOKSION OF CKI.TAIN 

 We find in the same way as before 



rf-y</r + 



dw/dg + ^ re- sin 27 = 



dtv/dr) -\- % TC' sinh 2f = 77 



OF S1IA1-TIXC. 



= 0. 



/8, - a < < + a 





(24). 



There is, however, in the present case, a further condition. For, referring to 

 figs. 6-8, when we cross the line SB,B>, S being that focus of the conies which is 

 inside the section, 77 is continuous, passing through the value 0, but f is discontinuous 

 and changes from positive to negative. We have to ensure that the function of 

 f and 77, which shall represent iv, shall l>e continuous in crossing 77 = 0, and that its 

 space-differential coefficients shall also be continuous. The latter condition implies 

 that dtv/dr) must change sign ; this l>ing so, all the conditions will l>e satisfied. 

 Let us write 



., sinh L' sin 2 if . 

 w = ire' - 



conditions (24) then 



. . (25). 



If we now assume 



= when f=*, < 7j < /8 



cut* 2/9 

 - 

 cosh 2 



when TJ = ft, 



I _ :==-)rfnhS8B o 



cosh 2mJ 



= S A,, sinh 



11 = 



sin 



then the conditions of continuity and the first of (25) are identically patisfied. 

 A B is found from the second equation of (25) in the same way as in 4. 



-1)" (conh 2 - COB 2/9) 



A B =-- 



7r(2n 



And thus 



sinh 2f sin 2i) n m 



if = ire - IGnra- (cosh 2* cos 26) 



* IPOSIl 'in V ' 



icosh 2 



X 2 



sin 



L'x ) 



2a 



l)ir[7r t (2n + 1)* 4 lC']coh 



VOL. 



. A. 



