CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 471 

 where K is a real constant, and 



........ <> 



Clearly the formula (96) does not become definite until a functional relation is 

 specified between x and 6. There are two ways in which the attempt may be made 

 to utilise the formula. One is the purely empirical device of assigning arbitrarily 

 such relations between x an d $ as would make the definite integral susceptible of 

 precise evaluation, and then ascertaining the corresponding configurations of fixed 

 and free boundaries. The other is to seek the particular functional relation which 

 shall correspond to an arbitrarily assigned form of fixed boundary. 



If x =/(0), ^ takes the form 



wherein it may be noted that the logarithmic infinity, which would correspond to 

 = w in the particular case of w real, does not destroy the convergence of the 

 integral. On substitution of this in (95), dz being represented by cfo exp ($x), it 

 appears that when w is real and between zero and a, 



x = _ p7r _ [j(8)dO=f(w), 



Jw 



f(a) being taken equal to >TT, and 



7 TT- 7 / / T I* V 



as = K aw -n n- exp - 



\a h w''/ TT 



Suppose the intrinsic equation to the curved part of the fixed boundary to he 

 prescribed, so that 



where F is a prescribed function. Then it appears that 



This relation specifies the property which the unknown function /(w) is required to 

 satisfy. It is an integral equation in/' (w), of by no means encouraging appearance. 



39. It is clear that the practically useful method of applying formula (97) is to 

 assign a convenient form to/(0) and to enquire what sort of problems can thereby be 

 solved. To this end it is convenient to separate ^ 29 or V^ in the first instance from 

 any transformation such as (95) and to study it as a double curve-factor having two 

 contiguous linear ranges, namely from oo to 0, and from to a, with the special 

 property that on the former of these ranges it has its modulus independent of ^v. 



VOL. ccxv. A. 3 R 



