CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 473 

 A type of ^ 13 is 



which has the linear range co < w < a K, angular range %-ir, and modulus, in the 

 range, (a w) 1 ' 2 . 



Thus <$/ Md * has angular range ^TT/ (/<) d/c and modulus (a iv)' l ' f ' K) ' lf . 



It follows that 



Lira "iT ' + i0 



has a double range of curvilinearity consisting of (i) --Gc<?r<0, on which the 

 modulus is Lim II ( ')'' 2/( " )l '", and (ii) < w < a e, on which the modulus lias a 

 less simple form. The angular range is Lim ^ir2,f(ic) die. 

 In general the limit form is 



f" f 



with angular range ^TT ./(/c)c&c an d modulus (a ')' /! J- /( <( ". 



f" 

 If an angular range pw be desired f must be chosen so that I J ( K ) '^ K = J>- 



The transformation 



dz = (w-a^^dw ......... (102) 



would give a configuration different from fig. 14 in this respect that the fixed 

 boundary would be straight from w = a to w = a e, and would then round smoothly 

 into a curve from w = a e to tv = 0. 



It is to be remarked that < @ K is not a curve-factor for K = 0, though it is so for any 

 positive value of K however small. This might seem to preclude the putting of e = 

 in formula (101 ). However there is no real difficulty, for the subject of integration 

 has no discontinuity at K = 0, provided due precautions have been taken in the choice 

 of/; and in any case the integral with lower limit zero is concerned with (amongst 

 others) vanishingly small values of K, but not with the actual value K = 0. If the 

 form of ^, 2 for e = be denoted by *$.&, the transformation 



gives a configuration like that of fig. 14. 



A simple example is afforded by giving to f( K ) the value 2pa~ l . It is readily 

 verified that 



(a 

 log {K' 

 . 6 



d K 



3 R 2 



