276 ON THE RECENT PROGRESS OF ANALYSIS. 



since if it be composite the argument of the circular or elliptic 

 function may first be divided by one of the factors of n, and the 

 result thus got by another, and so on. Thus setting aside the 

 particular case of n = 2, we shall have to consider, in order to 

 determine sin a or </>a, an algebraical equation of the nth or (n*)th 

 degree respectively. 



In consequence of the periodicity of sin a, the roots of the 

 equation in sin a admit of being expressed in a transcendental 



form; they are all included in the formula sin (a + -f\ , in 



which p is integral, and which therefore admits only n different 

 values. 



But elliptic functions are doubly periodic, and therefore the 

 roots of the equation in <a are expressible by a formula 

 analogous to the one just written, but which involves two in- 

 determinate integers corresponding to the two periodicities of the 

 function, just as p does to the single periodicity 2?r. Giving 

 all possible values to these integers, we get n 2 different values 

 for the formula. 



The question now is, how are we to pass from the transcen- 

 dental representation of these roots to their algebraical ex- 

 pression ? Or, in other words, how are the relations among the 

 roots deducible from the circumstance of their being all included 

 in the same formula, to be made available in effecting the solu- 

 tion of the algebraical equation ? 



The answer to this question is to be found in the following 

 principle : that if ^u be such a rational function of u that 



x, y, . . . z being the roots of an algebraical equation, then any 

 of these quantities may be expressed in terms of the coefficients 

 of the equation. This follows at once from the consideration 

 that we shall have 



IJL being the number of the roots x,y,...z. For the sum within 

 the bracket being a rational and symmetrical function of the 

 roots, is necessarily expressible in the coefficients of the equation, 

 and the same is therefore of course true of %#, or of any of the 

 other quantities to which it is equal. 



