CONFOEMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 467 



The process would be to take a real variable Q representing k r and to put f(6)d9 

 for n r , f(0) being a real function not involving w. It would generally be legitimate 

 to replace the limit of the product in formula (88) by the exponential of the limit of 

 the sum of the logarithms of the various factors, which last limit is an integral. 

 There results the very general curve-factor 



w-6+(i V 2 -cT'}d6, ..... (89) 



in which there is no d priori restriction of f(6) to continuity, beyond the exclusion 

 of such infinities as would prevent convergence of the formula. There may, for 

 example, be such infinities, for particular values of 9, as would make *$.& include as 

 factors definite powers of the corresponding forms of ^ 8 . 

 For values of w on the linear range the modulus of ^ 28 is 



- 



and the vector angle 



P f(d) tan- 1 {(J-w^'Kw-e) }d6. 



J -c 



The angular range is 



f(e)de. 



LIQUID MOTIONS WITH FREE STREAM-LINKS. 



34. In a liquid motion the characteristic of a stream-line which is free, or is a 

 possible line of discontinuity in the motion, is that along it the resultant velocity q 

 is constant. Now q- 1 = \ dz/dw , and therefore a transformation of the type 

 dz/dw =f(w) will give as part of the boundary a possible free stream-line if there 

 is a part of the range w real for which f(w) \ is constant. 



The simplest example of this is presented by the curve-factor ^ ; since the modulus 

 of this, within its linear range, is constant, a transformation in which /() consists 

 solely of a power of < @ l gives a free stream-line as part of the boundary. In fact, the 



transformation 



dz = ^{w+(w 3 -c 2 p}"dw (90) 



gives a configuration of the general character indicated by fig. 1, with this special 

 feature that on the curved part of the boundary there indicated the velocity is 

 constant. Thus, so far as the ordinary theory of discontinuous fluid motion is 

 concerned, the curved part of the boundary may be a free stream-line. 



35. Other cases of free stream-lines across finite gaps in boundaries which are 

 otherwise rectilineal may be built up from curve-factors of the types ^ and #. 

 These being taken in the forms 



