CONFORMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 461 



so that it has an angular range TT corresponding to the linear range to oo . The 

 order at infinity being unity there are .no imaginary zeros in the relevant region, 

 and so $",<, is a confbrmal curve-factor. It has something of the character of a double 

 curve-factor. 



DOUBLE CURVE-FACTORS. 

 27. An example of a double curve-factor is afforded by the function 



-6)} l S ..... (70) 



where b > a, > 0. The form of this must be considered for different parts of the 

 axis of w real. 



For w > b the function is real ; moreover w 2 ^ (a 3 + b 3 ) is positive, so %>. M has no 

 zero in this part of the axis of w real. 



For b > w > a the form is 



of which the real part changes from positive to negative as ir diminishes through the 

 value +{i (a s +6 2 )} 1 ''. The vector angle accordingly ranges from zero to tr. 

 For a > w > a the form is 



of which both terms are real and negative. 

 For a > w > b the form is 



of which the first term changes from negative to positive as w diminishes through 

 the value -{ (a s + 6 2 )} 1/2 . The vector angle increases from TT to 27r. 

 For w <b the form is 



of which both terms are positive. 



Thus the factor corresponds to two curves separated by a straight line, and over 

 the whole range from + b to -b the angular amplitude is 2*-. The order at infinity is 

 2, so there are no imaginary zeros in the relevant region. It appears therefore that 



^20 is a curve-factor. 



28. It is interesting to enquire what sort of a configuration is given by a trans- 

 formation which involves # and also Schwarzian factors introducing corners at 

 points a, b. Such a transformation is 



, 



= 



dw (71) 



' 



