CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 481 



corner, where x undergoes finite change, say pir, while 6 is at a standstill say at a, 

 contributes to the integral an amount p-ir log (w a), and so leads to the ordinary 

 Schwarzian or corner factor. The curved portions of the boundary contribute 

 distinct curve-factors of the type ^ 41 . 



For any one continuously curved stretch of the boundary, say a < w < b, let / t (0) 

 be the suitable expression for x/^, and let Fj (w) be the modulus of all that part of 

 the right-hand side of formula (123) which corresponds to values of outside 

 the range a < Q < b. Then the condition that /i has to satisfy is of the type 



. . . (124) 



To other curved parts of the boundary correspond other functions, / 2 , /-,,..., &c., 

 which satisfy similar conditions involving F 2 , F 3 , ... , &c., but F! depends on f 2 ,fz, , 

 and F 2 depends on/,, f :t , ... , so that the complete expression of the conditions which 

 the/'s must satisfy is a complicated set of simultaneous integral equations with the 

 function log | wQ \ as kernel. 



48. Free Stream-lines. As an alternative to the formulation of the previous 

 article it may be assumed that along the curve x = ^( s )> an d s = g (w), where g is 

 the unknown function. Then 



-O dedw, 



{ .'a 



so that 



= o, .... (125) 



an integral equation (or equations) in g'. 



The characteristic property of a free stream-line is that along it the velocity 

 is constant, say unity, so that w = s and g' = 1. Thus if one portion of the 

 boundary, instead of being of prescribed shape, is to be a free stream-line, the 

 corresponding function \js must satisfy the condition 



-8\d6 = Q, ..... (126) 



an integral equation in >// with kernel \og\w 0\. 



The solution of this equation for the case in which the fixed boundary is a 

 rectilineal polygon is derivable from the results of 37. 



TRANSFORMATIONS INVOLVING BOTH VARIABLES EXPLICITLY. 

 49. The study of conformal representation by means of transformations of the type 



would obviously present even greater difficulty than the kind of transformation 



3 S 2 



