460 DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



and it appears that when w is negative 



(66) 



The utility of a knowledge of the moduli of curve-factors will be exemplified later. 

 24a. Curve- Factors involving Powers other than the Square Root. A more general 

 type of curve-factor having semi-infinite range is obtainable as follows. Consider 



.......... (67) 



where |- > a > 0, and both b and b" are positive. This has no real zeros, since both 

 terms are positive when w > 0. The linear range of curvilinearity is from zero to 

 oo, and on this range w" is represented by ( w) a exp (ionr), so that the vector 



angle x of ^ 17 is 



w) a sin 

 - - - 



= tail 



- - - 

 \b a +(-w) a cos 



Tlie denominator in this expression cannot vanish, so, as w decreases from zero, 

 X increases to cnr ; thus the angular range is aw. The order at infinity is a, so there 

 are no imaginary zeros in the relevant region. Hence ^ 17 is a curve-factor. 



The case of 1 > a > ^ has been excluded because it would give an angular range TT, 

 and so introduce imaginary zeros into < $ ll . 



25. Consider also 



# ]8 = > + // + & l -* ......... (68) 



where k' > 0, 4- > a > 0, and both b and b l ~" are positive. This has no real zeros 

 since all the terms are positive when ir > 0. On the range of curvilinearity the 



vector angle x is 



b l ~ a ( w) a sin 

 X = tan l 



l - a 



The denominator in this expression is negative for w = < and positive for w =k' 

 and w = 0, and so has a zero for a value less than k'. Therefore as w decreases 

 from zero x increases through |TT to TT, and the angular range is TT. The order at 

 infinity is unity, so there are no imaginary zeros in the relevant region. Hence ^ 18 

 is a conformal curve-factor. 

 26. The function 



........ (69) 



wherein X, /*, v , and a are all positive, is real for w real and positive. For > w > a 

 it has the vector angle 



tan- 1 [^{(w + a) (-;)}'/= + (-w^/X, 



and for w < a the vector angle 



tan- 1 -w 



