REPORT ON CERTAIN BRANCHES OF ANALYSIS, 321 



which satisfy the congruence 



,r3 _ 1 = M X 7, 



- 1 + 7 + ^ — 3 + 7 



whose modukis is 7; are expressed by 



2 



and ~ — — , which arise from adding 7 to the 



parts without and beneath the radical sign. 



The principle of this transition from the root of the equation 

 to that of the congruence is sufficiently simple. We consider 

 the roots of x" — 1 = as resulting from the expression for 

 those of the congruence or" — 1 = M (/>), when M = ; and 

 we thus are enabled to infer, in as much as M (/>), its multiples 

 and powers, are involved in those formulse, whether without 

 or beneath the radicals, and disappear, therefore, when M = 0, 

 that some such multiples, to be determined by trial, or other- 

 wise, are to be added when M (/?) is restored, or when 1 is 

 replaced by 1 + M (p). When the congruence admits of in- 

 tegral values of jr, which are less than p, then they can be found 

 by trial : when no such integral values exist, then, amongst the 

 irrational values which thus arise, those values will present them- 

 selves which will satisfy the congruence algebraically, though 

 they can only be ascertained by a tentative process. 



The equation of Fermat, 



^p-' — 1 = M (^), 



where /> is a prime number, will be satisfied by all the natural 

 numbers 1, 2, 3, . . as far as (^ — 1) : and it follows, therefore, 

 that all the rational roots of the equation 



a7» — 1 = M (j9) 



will be common to the equation 



x^-i - 1 = M (^), 



the number of them being equal to {d), the greatest common 

 divisor of w and oi p — \. If <^ be 1, then all the roots except 

 1 are irrational. If we suppose the equation to be 



tp - 1 = M (p), 



then all the roots will be equal to each other and to 1. It is 

 unnecessary, however, to enter upon the further examination 

 of such cases, which are developed with great care and sin- 

 gular ingenuity in the memoir referred to. 



These views of Poinsot are chiefly interesting and valuable as 

 connecting the theory of indeterminate with that of ordinary 



1833. Y 



