ÖFVBRSIGT AF K. VBTENSK.-AKAD. FÖRHANDLINGAR 1901, N:0 9. 695 



and therefore by passing to the limit 



= 9„ (36) 



which is the result in (I") before and in another manner given 

 by Lagrange and Laplace. 



From the value of M given in (21') we see, that if the 

 coefficients in the equation (10) are altered, while the exponents 

 of z remain unchanged, then M only could be altered by means 

 of an alteration of the difference, /' — I", between the number 

 of roots, whose modulse are respectively less and greater than 

 unity. 



We assume, that all the roots, the roots with a modulus 

 equal to unity included, in respect to the raagnitude of their 

 modulus can be divided in s groups, the number within each of 

 these groups being assumed to be 



(Ti, 02, . . ., Os, 

 so that 



ö-j -\- O^ + . . . + Gs = f^x- ^^^0 



The modulas of the roots within the same groups we denote with 

 and assume that they are subject to the conditions 



ka,l>k<.2l- •>|?o.|- (38) 



New let a be an arbitrary complex quantity and let the 

 coefficients in the right number of (6) be changed into 



^ja»»i , A^a"^^ , . . . , Ana"'n , (39') 



whereby thus the equation (10) is changed into 



^,a^'M-"' + A^af'iiif^^ + . . . + A„ = O . (39) 



Accordingly this equation by (11) possesses the roots 



ß m o w, p in 



a a a 



If I a I difFers from unity, we see that none of these roots 

 can have a modulus equal to unity. That is: the equation (39) 



Öfvers. af K. Vet.-Akad. Förh. 1901. Arg. 58. N:o 9. 2 



