ÖFVERSIGT AF K. VETENSK.-AKAD. FÖRHANDLINGAR 1901, NIO 9. 



Moreover we assume these convergents to be so accurate 

 that conformably to (2) the unequalities 



mj > rwo • • • > ^'^n (5) 



also take place. 



Further we assume 



r=l 



so that, when the convergents (4) converge against the values 

 (3), the modulus r\ and the argument 6^ converge against the 

 modulus r and the argument e, and the equation (6) thus in the 

 limit coincides with the equation (1). 



Putting 



t 



i — 



z = e m 



the right member of the equation (6) is changed into the rational 

 polynomial in respect to 2;: 



= A,Z"^n l^^i +^'z^^2 + ,,. + ^\^ (7) 



where we have assumed 



A*l = ^1 '^n , /"o = m^ — m^ , . . . , f.ln~l = 'Wn-l — w« , (8) 



from Avhich it further follows according to (5). 



l-l^ > ^<2 • • • > f-^n ■ ■ (9) 



The expression within brackets in (7) is thus a rational 

 polynomial of the degree (.i^. 

 As the equation 



A^u^'i + A^uf'i + . . . + A,i = (10) 



can not have any root equal to zero, all the coéfficients A being 

 assumed significative, its roots may be denoted by 



