REPORT ON CERTAIN BRANCHES OF ANALYSIS. 317 



celebrated Gauss, in 1801, gave an immense extension to our 



knowledge of the theory and solution of such binomial equa- 



x"—l 

 tions. It was well known that the roots of the equation — ; — r- =0, 



where w is a prime number, could be expressed by the terms of 

 the series 



r + r'^ -\- r^ + . . . r"~^, 



where r represented any root whatever of the equation, and 

 where, consequently, the first term r might be replaced by any 

 term of the series. But in this form of the roots there is pre- 

 sented no means of distributing them into cyclical periods, nor 

 even of ascei'taining the existence of such periods or of determin- 

 ing their laws. It was the happy substitution of a geometrical 

 series formed by the successive powers of a primitive root* oin, 

 in place of the arithmetical series of natural numbers, as the in- 

 dices of r, which enabled him to exhibit not merely all the dif- 



X 



- 1 



ferent roots of the equation j = 0, but which also made 



manifest the cyclical periods which existed amongst them. 

 Thus, if a was a primitive root of n, and n — I — mk, then in 

 the series 



**, r , r , r , . , , r , . , . r , 



the m successive series which are formed by the selection of 

 every k^^ term, beginning with the first, the second, the third, 

 and so on successively, or the k successive series which are 

 formed in a similar manner by the selection of every m^^ term, 

 are periodical ; and if the number m or k of terms in one of 

 those periods be a composite number, they will further admit of 

 resolutions into periods in the same manner with the complete 

 series of roots of the equation. The terms of such periods will 

 be reproduced in the same order, if they are produced to any 

 extent according to the same law, it being understood that the 

 multiples of n which are included in the indices which succes- 

 sively arise, are rejected, for the purpose of exhibiting their 

 values and their laws of formation in the most simple and ob- 

 vious form. If two or more periods also are multiplied together, 

 the product will be composed of complete periods or of 1 , or of 

 multiples of them, the rules for whose determination are easily 



• There are as' many primitive roots of n as there are numbers less than 

 n — 1 ■which are prime to it. Euler, who first noticed such primitive roots 

 as determined by Fermat's theorem, determined them by an empirical pro- 

 cess. Mr. Ivory, in his admirable article on Equations, in the Supplement 

 to the Encyclopccdia Britannka, has given a rule for finding them directly. 



