CONFORMAL TRANSFOEMATION TO PROBLEMS IN HYDRODYNAMICS. 465 

 in which fi> > 2 . It can be readily verified that 



I'l 



25 





Consequently both these types of inflexional curve-factors are equivalent to simple 

 fractions having curve-factors of the type of ^ in numerator and denominator. 

 In the case of a semi-infinite linear range it is seen that 



'It = 



1 ') ) . . . (84) 



so that ^ ]6 is equivalent to the product of two curve-factors of the type of ^ 13 . Also 

 (with the notation of article 24) 



_ w -k+ fc/r) 1 " _/''- + 1 (;''- 

 w-0 V'+w 1 '' 1 



in which fraction both numerator and denominator are of tlie type of $*,;,. 



32. Of the curve-factors which have been studied some are such that, for real 

 values of w which render them complex, the squares of their moduli have real factors ; 

 others have not this property. To the latter class the theorems of the previous 

 article do not apply, but as regards the former class the results suggest a useful 

 method of generalisation. Within the limits of this class of curve-factors it seems 

 possible to regard the types ^ 6 and "^ as fundamental for the finite linear range c 

 to + c, and the type ^ 13 for the semi-infinite range -- o to zero. Bearing in mind 

 that any power of a curve-factor, whether positive or negative, is itself a curve-factor, 

 one may take any number of curve-factors of the types ^ and < $.^, raise each to any 

 power, and multiply all together, so producing a resultant curve-factor for the 

 assigned finite linear range. In the same way one may take the product of arbitrary 

 powers of any number of curve-factors of the type <$ w , so getting a resultant curve- 

 factor for the assigned semi-infinite range. These curve-factors have a quite arbitrary 

 number of adjustable parameters, and so correspond to an endless multiplicity of 

 curves in the plane of z. 



g 



If, for example, the ship problem already discussed in article 1 5 be again considered, 

 it is legitimate to take a set of parameters k l} k. 2 , ... k r , ... intermediate in value 

 between c and +c, and a set of corresponding indices n^ n 3 , ... n r , ... ; also a set 

 of parameters l l} 1 2 , ...I,, ... greater than c or less than c, and a set of corresponding 

 indices m lt m 2 , ... m,, ... . These combine to give the curve- factor 



c 3 )' l >}'U{l.-w+(iv 3 -c>) ll '}'; .... (86) 

 3 Q 2 



