222 Proceedings of Philosophical Societies. [Sept. 



of observation, and partly to the inequalities which cause the 

 variation of the distance. 



The author then endeavours to find the influence which these 

 different inequalities may have on the longitude and latitude of 

 the lunar spots, seen from the centre of that satellite. He gives 

 an analytical expression of it, which must be compared with 

 observations, in order that the differences between the momenta 

 of inertia of the lunar spheroid may be found from it, as well as 

 the two constant quantities relative to the spot that is observed. 

 This comparison is assigned to M. Nicollet, and he proposes to 

 publish the results as soon as any satisfactory ones have been 

 obtained. 



Upon the Application of Algebra to the Theory of Numbers, 

 by M. Poinsot. — In this memoir, the author has principally in 

 view a demonstration of the general theorem which he has o-iven 

 relative to the algebraical expression of the imaginary roots of 

 unity, with some remarkable applications which he had indicated 

 in his preceding researches into algebra and the theory of num- 

 bers. To give a general idea of this theorem, let us consider the 

 indeterminate binomial equation a?" — 1 = M p, in which the 

 second term M p expresses any multiple of a prime number p, 

 and n any prime exponent that is a divisor of (p — 1) in order 

 that the equation may have n roots or solutions in whole numbers 

 inferior to p. 



The author shows that if in the place of this equation there be 

 taken the determinate binomial equation at" — 1=0, and it is 

 resolved, the algebraical expression of its roots which, except 

 unity, are entirely imaginary, will be the analytical expression of 

 the different whole numbers which resolve the equation x n — 1 

 = M p ; that is to say, by adding the proper multiples of p to 

 the numbers which are under the radicals of this formula, the 

 imaginary and irrational ones will disappear, all the operations 

 pointed out are rendered capable of being perfectly executed, 

 whole numbers which resolve the proposition will be obtained, 

 and the formula will only express those very numbers. 



The author establishes this theory for every case of the expo- 

 nent «, simple or compound, prime or not, with (p — 1). 



When the exponent n is an exact divisor of (p — 1), the 

 equation x n — 1 = M p has always n roots in whole numbers, 

 as is easily to be deduced from the famous theorem of Fermat. 

 But if n does not divide the member (p — 1), the equation has 

 only a single root, or entire solution, which is unity, and all the 

 others are always impossible or irrational quantities. Neverthe- 

 less, the imaginary formula, which expresses the «th root of unity, 

 is still the analytical expression of even these impossible roots. 

 This expression is, therefore, as perfect as those of imaginary 

 quantities in analysis ; that is to say, it may be employed with- 

 out any fear in analysis ; and if by any combination of similar 

 values, the irrational quantities should happen to be destroyed, 



