p-n jrXCTIOXS /.V SKMhOXnrCJORS 485 



are monotonically iiKTeasin<^ luiulions of .1,, . This may be seen by con- 

 sidering -v = .v„ . For .1 a sufficiently small, the value of Ji{xa , Ag) and u'{xa , Aj 

 are given simply by (A7.4). For larger values of .la, an exact integral will be 

 required. It is evident, however, that all solutions of the form (A7.4) are 

 related simply by translation for .v < Xn . Hence increasing An is simply 

 equivalent to integrating (A7.1) to larger values of .v and it is evident that 

 this increases n and u' monotonically. It may be verit'ied that for a sulliciently 

 large .!« the solution becomes infinite at .v„ so that i((x a, Aa) u'{xa , Aa) both 

 vary monotonically and continuously from — ^- to -{- x as .la varies from 

 negative to positive values. \\'e shall refer to this property of u(xa , A„), 

 ii'{xa , A a) as Pi . 



We next wish to show that h{xi , .!„), u'{xi , .4a) has the property Fi for 

 \alues of .Vi > .v„ . To prove this we note that if for any .vi , m(.Vi , .4a) and 

 /<'(.vi , A a) are finite, the solution may be integrated somewhat further to 

 obtain «(.V2 , .4a), n'{x-> , .4a) for .V2 > Xi . From equation (.^.7.1) it is evident 

 that an increase in either «(.Vi , a) or w'(.Vi , a) will result in an increase in 

 d'-u dx- in the interval Xi < x < .Vo so that u and u' at x^ are monotonically 

 increasing functions of u and n' at Xi . Hence if ti and u' at .Vi have the 

 I)roperty Fi , so do u and u' at .v; . By extending this argument we conclude 

 that H and ;/' at any value of .i- have the property Pi . (A rigorous proof 

 can easily be completed along these lines provided that |/(.v) ] is finite.) 



Similarly it may be shown, starting from (.•\7.5), that u{x, Ab) is a mono- 

 tonically increasing function of .1,, and h'{x, .1?,) is a monotonically decreasing 

 function of Ai, . 



In order to have a solution satisfying (.^7.4) and (.A7.5) we must have, 

 for any selected point .v, 



n{x, Ac) = h{x, Ab) (A7.6) 



;/'(.v, A,) = u'(x, Ab) (A7.7) 



Xow as the equation u(x, .!„) = n{x, Ab) varies from — --c to + x , ;/(.v, .4^) 

 varies from — x to + x and u'(x, Ab) varies from + --c to — x , monotoni- 

 cally and continuously. Hence there is one and only one solution of (.'\7.1) 

 satisfying (.^7.4) and (.\7.5). 



In order to verify that the solutions discussed in Section 2 are correct for 

 large and for small A', we show schematically in Fig. Al the solution for a 

 representative A' as a dashed line together with the curve u = Uoiy) = sinh" 

 y. In terms of 7^0 , equation (2.16) becomes 



—r — -— (sinh H — sinh ih). (A7.8) 



dy- A- 



