472 DR. J. G. LEATHEM ON SOME APPLICATIONS OF 



It is convenient to remove the logarithm from the integral by integration by parts, 



thus 



By way of illustration, let /(0) = X0 1 ' 3 , then indicating the particular case by a 

 fresh suffix, 



TT IQO- ^. n = X0'' a log( ,1 irj + Xw 1 " log \W( 



and 



# ;ii = ["'';;+'j^( w -j^ (100) 



For v real and greater than a, ^ is clearly real. For a > n~ > it takes the 

 form 



.' '/-.. i r /.V Ar '' fe l a _ >r A " ' i/., i;., 



and its vector angle ranges from zero for = a to Xa"' for ?r = 0. 

 For real and less than xero it takes the form 



A ''-L^'>_ , 



w - ik/i ' 



which has modulus unity, and vector angle 



\x~ 1 ^2o 1 '*tan~ 1 ( -) +( w) 1 ' 2 log 



a 



wliich ranges from Xa 1 - for ?r = to zero for w = -co. 



The assigning of different values to the parameter X is equivalent to taking 

 different powers of the factor corresponding to X = -K. 



If ^ 3 , be employed in the transformation (95) the form of the curved part of the 

 fixed boundary depends on both the parameters X and p. 



40. There is another method, slightly different from that just discussed, of building 

 up a product whose limit is a curve-factor of a type suitable for dealing with problems 

 of the class represented by fig. 14. In 38 important factors of the product were 

 curve-factors having all the same semi-infinite linear range of curvilinearity but 

 different moduli ; in the present instance curve-factors having different semi-infinite 

 linear ranges, but the same modulus, are multiplied together. 



