Leathem — On Two-Dimensional Fluid Motion. 19 



The passage to limit yields the formula 



-mr^^m^ ™ 



The integral in the second line is not, of course, definite until assumption is 

 made of a functional relation between k and A, that is between the integrated 

 How up to a point of the boundary and the direction of the tangent at that 

 point. The form of this function and the form of the fixed boundary are 

 interdependent, so that if either is assigned the other is determinate. 

 The corresponding geometrical relation is 



^-{^rM'-f^TV 



'«•<- 



and this, together with the field relation, with the hypothesis of an adjustable 

 functional relation between k and A, constitutes the general formulation of 

 the hydrodynamical problem. But the generality of the formulation has yet 

 to be proved, and this must now be done. 



Demonstration of Generality. 



8. The corner of angle 2^>7r at the vertex being assumed to be the only 

 corner on the obstacle, though others could easily be allowed for if necessary, 

 the theorem to be proved is this: — Any curve-factor whatever, A (£), which 

 has the property that its modulus is constant over those parts of the real 

 axis of £ for which Z? > a % , and which has no zeros or infinities on that axis 

 save an infinity at c of the form (£ - c)~ !p , is capable of being expressed as 

 in formula (10). Without any real loss of generality, one may postulate 

 further that £(%) tends to a real limit, K, for t ->- + od . 



It is convenient to introduce a new function defined by the relation 



F[k,Z) 



*_1- 



IK 





- kl, + a- - i (a- - K-j- (c -cry ( i - 1 



which, for k real and | k | < ". is a curve- factor in Z whose limit value for 

 £-$>-» is unity, with constant modulus (unity) over the proper ranges 



R.I. A. PROC, VOL. XXXIV, SHOT, A. [1] 



