CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 479 



integrations, and so generating curve-factors whose geometrical significance in typical 

 transformations can be examined. As the geometrical interpretation has to do with 

 real values of w, it is to be remarked that for w real and b > w> a 



-[ Iog(w-8)f'(et)d6=-\ b log\w-6\j'(e)d6+iir{f(w)-f(b)}, . (114) 



J a J a 



so that the vector angle of ^ 41 is >r{f(w)f(b)} and its modulus 



expj- "iog\w-e\f'(e)de}. 



I Ja 



A simple example is got by taking/(0) = p9 + q, where p and q are constants, so 

 that dx = TrpdQ. The argument of the exponential in the curve-factor is 



= p{ab+ (wa)log(wa)(wb) \og(wb)}. 



After omission of a constant factor this yields the curve-factor 



^ 42 = (,r-rt)><<--> (,,<-/,) -*<'<'-" (115) 



It is readily seen that the angular range of ^ ia is irp(b <i), and that for w great 

 it approximates to the form e (b ~" } w p(l '-"\ so that there is a definite order at infinity, 

 namely jo (6 a). The curve-factor does not tend to zero as w approaches a or b. 

 By putting f(0) = 9 2 one gets the curve-factor 



^ 4:j = (v f />) w; -"- 2) (7t'-) ( '" 2 -" J) e 1Ma -'' )( " + '' +2 " ;) , (116) 



with angular range 7r(b 2 -) and order at infinity (b 2 2 ). 



There would probably be little difficulty in finding a considerable number of forms 

 of f(6), which would permit of evaluation of the integral in formula (113) and so 

 yield types of curve-factor. 



46. The problem of formulating the condition which f(6} must satisfy, in order 

 that the resulting curve-factor (in combination with suitable Schwarzian factors) may 

 be applicable to a prescribed curve, may be exemplified by taking the case of the 

 doubly-pointed ship. Suppose a configuration like fig. 3 has to be dealt with, the 

 form of the curve in the linear range c < w < c being prescribed. The trans- 

 formation will be of the form 



-f f(e)]og(v-e)de\dip,. . . (117) 



and the question is what is required of f(0) in order that the curve corresponding to 

 c < w < c may be as prescribed. 



VOL. CCXV. A. 3 S 



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