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



10. The form of ^ 6 suggests another possible curve-factor, 



^ 7 = (;-&) cosh a +(w'-c') 1/!I sinh a (14) 



Any zeros which this may have must be zeros of 



# 7 # 7 = (w-k cosh 2 ) 2 +sinh 2 a (c 2 -F cosh 2 a) (15) 



If F cosh 2 a > c 2 , the zeros of ^ 7 ^ 7 are both real, and if c 2 > /P, < $ 1 cannot have 

 real zeros. Hence ^ 7 is a curve-factor if c 2 > k 2 > c 2 sech 2 a. 



If F cosh 2 a < c 2 , which involves a fortiori that P < c 2 , then ^ 7 ^ 7 has imaginary 

 zeros, and the question arises whether ^ 7 can have any imaginary zeros at points on 

 the relevant side of the axis of w real. This may be tested by means of the theorem 

 that, if a function f(w) be free from infinities in a given region, the sum of the 



orders of all its zeros in that region is equal to (l/27rt) \f'(w)/f(w)dw taken round 



the boundary ; a simpler enunciation of the theorem is that the sum of the orders of 

 all the zeros of ,/'("') ' ln the region is 6/2-Tr, where B is the algebraic sum of all the 

 changes, abrupt or gradual, which take place in the vector angle of f(w) as w makes 

 a complete circuit of the boundary in the positive sense. 



In the present application ^ is put for /(;), the region is the half- plane of w on 

 the positive side of the real axis, and the boundary consists, in the main, of the whole 

 of the real axis and a semi-circle of radius 11 which tends to indefinite greatness. 

 The points w = +c, being branch points, have to be avoided by semi-circular detours 

 of infinitesimal radius on the positive side of the axis. If there were zeros on the 

 axis they would have to be avoided by similar detours, but in this case there are 

 none. The points w c not being zeros or infinities, the detours round them do 

 not result in any alteration in the vector angle of <@-. From - oo to c this vector 

 angle is -w ; from c to +c it continually diminishes (because c > k > c) from -w to 

 zero; from -t-c to + co it is zero; along the semi-circle from + oo round to oo it 

 increases from zero to TT. Hence 9 = -TT + TT = 0, and so ^ has no zeros in the 

 relevant half- plane. 



Therefore, if k 3 < c 2 sech 2 a, <$ is a curve-factor. 



11. Relation let-ween Angular Range and Order at Infinity. In the test for the 

 presence or absence of imaginary zeros employed in the previous article it will be 

 noticed that the sum 6 is made up of two parts, one corresponding to the linear 

 range and equal to minus the angular range, the other corresponding to the infinite 

 semi-circle and equal to T multiplied by the order at infinity of the curve-factor. 

 From this it is seen that if an expression of the type 



f(w)+g(w)(w'-c?) 11 ' (16) 



is free from infinities and real zeros, and has its angular range equal to TT times its 

 order at infinity, it will also be free from imaginary zeros in the relevant region, and 

 is a conformal curve-factor. 



