Lkathiom — On Two- Dimensional Fluid Motion. 15 



4. It is convenient to put u + iv = q exp (iw), so that q is the resultant 

 velocity and rc its direction-angle, and to notice' that dw/dz = u - iv. The 

 characteristic of a stream-line that is free is that, along it, the resultant 

 velocity q or | dw/dz | is constant. 



If the geometrical relation be 



dz = g a (i) dz, (2) 



where (?„(£) is a curve -factor, the constancy of <? involves that of | £ (£)dZ/dw |, 

 so that, if w= §V(£-cy, | ^ (O/(? _ c ) I is constant. Along a free stream- 

 line which extends to infinity the constant value of q must be V, and therefore 

 the constant value of | £»(£)/(£ - c) | is unity. 



If, as in the case of symmetry, w = t,, then | Go%) | is constant along a 

 free stream-line. 



General Fokmulae. 



5. A field relation and a geometrical relation of the forms of equations 

 (1) and (2) respectively having been assumed, the problem of formulation is 

 reduced to that of rinding a suitable general formula for (p (Z). 



The requirements which (r'„(Z) has to satisfy are : - (i) that it shall give a 

 conformal relation between the relevant regions of the planes of z and £, 

 which is equivalent to saying that it must be a curve-factor, not necessarily 

 pure ((?, § 1), whose angular range is it, and (ii) that all along the parts of 

 the boundary corresponding to free stream-lines, namely the parts where Z, is 

 real and £ 2 > «-, the modulus of 6o(S)/(£ - c) shall be unity. 



In the case of symmetry there is a single relation, say 



dz = £ '(w) dw, (3) 



and what is required of Ga(w) is that it shall he a curve-factor of zero angular 

 range, whose linear range is the part of the real axis of w in which w is 

 positive, and such that the modulus of (/,'{w) is constant for w real and greater 

 than a. 



Whether for & T (£) or ff a \K), the general formula is a product or limit of a 

 product of curve-factors of suitable type and possibly also Schwarzian 

 corner-factors. But there is a considerable range of choice of the type 

 of sub-factor. In the case of a limit of a product it is convenient to 

 represent 6 T or (/„' as the exponential of a definite integral, making use 

 of equalities of the type 



« = S /(<c) 8k 



Li m n {0M) = Ex p 



b 



/( K ) log {0 (*)} <fc. 



