318 



connection between physical parameters and the error in the first approxi- 

 mation . 

 4* Thin-Plate Equations in the First Approximation 



In obtaining the solution in the first approximation the first 

 step is to solve Eqs. (18) and (19) for '^^> in terms of S with 

 the result: 



(0) 



Also 



^>-l-A"'*j:rTup^l 



s^.i^sr^JTriirr]. 



(23) 



(24) 



is obtained. Since having 3 ^^^ 3^ both positive is the only case of 

 interest here, only the positive sign in front of the radicals will be re- 

 tained henceforth. It can be shown that 'SS, /«)r possesses a saddle- 

 point singularity at the point. (0,1) in the r^ S, -plane and that the 

 orientation of the singularity is such that no integral curves except 

 5 =1 cross the 3, axis. It immediately follows from Eq. (18) that 

 O i: 5 ** 6 x/yTS . The actual solution of Eq, (23) (using the plus sign) 

 is 



with the limiting case, 



S,"' = 1 • (26) 



Solutions (25) and (26) satisfy the qualitative restrictions already stated. 



10 



