expressed by Serieses of periodic Terms. 165 



zero : but distinctions must be made with regard to the other 

 factor. When the value of % 



(2(1 -p))^ " 

 is either always finite or infinitely small, all the differentials 

 of X" will be infinitely small as assumed by M. Poisson, and 

 the equation (2.) will be proved. But the same equation will 

 not be proved if the limit of the same factor be either infi- 

 nitely great, or if it be a quantity that cannot be generally 

 determined, and of which it cannot be said that it is either 

 finite, or infinitely great. 



If f = 1, the tactor in question will be, 



V2{\--py 

 which has a finite value when u, or J" {6, ^) is a finite function 

 of cos 6, sin fl cos \J/, sin fl sin \J/. This readily follows from the 

 usual transformation of such expressions. The same factor 

 will be infinitely small when u'—u is divisible by (l—p)", n 

 being any positive integer. In all these cases the equation (2.) 

 is demonstrated. If n = 1, or if ti'—u be divisible by 1—p, 

 we fall upon the instance particularized by Laplace in the 

 eleventh book of the Mecaji. Celeste. 



But if zi, or y(fl, vf/), be not a finite function of cos 9, sin fl 

 cos \I/, sin 6 sin vj/, no determinate value can be assigned to the 

 factor 



u'—u 



(2(1—^))^ 



by any transformation ; and in all such cases the equation (2.) 

 is not, demonstrated. 



Lagrange has considered this subject in the 15th cahier of 

 the Journal de I'Ecole Polytechnique. His investigation pos- 

 sesses all the exactness and clearness and elegance which di- 

 stinguish the writings of this geometer. But the success of 

 his analysis demands thaty(9', ^') shall be a finite function of 

 cos 6', sin 6' sin vj/', sin 6' cos 4/'. For such functions his pro- 

 cess leads to a strict demonstration of the equation (2.): for 

 functions of a different description, the algebraic operations 

 fail, and no other conclusion can be drawn except that the 

 equation (2.) is not demonstrated. It is very remarkable that 

 the illustrious author has not noticed a distinction so obvious 

 and necessary. 

 July 26, 1836. James Ivory. 



• Tkcorie de la Cha/ettr, [>. 2\3. 



