Manchester Memoirs, Vol. xliii. (1899), No. T. 3 



element of the normal to ds, the positive directions of these 

 elements having the same relation to one another as the 

 positive directions of x and y, then + zi// being any 

 function of x-\-iy we have 



dn ~ ds'' ds~ dn ' ' '5^' 



Hence, integrating round a closed curve which does not 

 pass through any singular points, we find 



M'^-f' 



^ # • ■ . . (6), 

 and 



/S*=-A • ■ ■ w- 



As a particular case we may put 



.^ = logi?, ^p^Q . . . (8). 



where R, are defined by (3). Cauchy shows that the 

 value of ( 



I 



u 



taken round any circuit which does not include a zero of 

 f(z) is zero, and it follows from (6) that for such a circuit 



/ 



"-r-° . . . (,). 



From this we may infer the non-existence of a minimum 

 value of log 7? within any region not containing a zero 

 of /(.-). 



It follows also from (7) that the integral 



/ 



dv du 

 u — - v~-- 

 d?i dn J / s 



-ds . . . . (10). 



u' + v' 

 taken round any closed circuit which does not pass 



