396 Mr green, ON THE DETERMINATION OF THE 



on the property which a certain function F'* then possesses of satisfy- 

 ing a partial differential equation, whenever the law of the attraction 

 is inversely as any power n of the distance. For by a proper applica- 

 tion of this equation we may avoid all the difficulty usually presented 

 by the integrations, and at the same time find the required attrac- 

 tions when the density p is expressed by the product of two factors, 

 one of which is a simple algebraic quantity, and the remaining one 

 any rational and entire function of the rectangular co-ordinates of the 

 element to which p belongs. 



The original problem being thus brought completely within the pale 

 of analysis, is no longer confined as it were to the three dimensions of 

 space. In fact, p' may represent a function of any number s, of in- 

 dependent variables, each of which may be marked with an accent, in 

 order to distinguish this first system from another system of s analo- 

 gous and unaccented variables, to be afterwards noticed, and F' may 

 represent the value of a multiple integral of s dimensions, of which every 

 element is expressed by a fraction having for numerator the continued 

 product of p into the elements of all the accented variables, and for 

 denominator a quantity containing the whole of these, with the un- 

 accented ones also formed exactly on the model of the corresponding 

 one in the value of V belonging to the original problem. Supposing 

 now the auxiliary variable u is introduced, and the s integrations are 

 effected, then will the resulting value of ^ be a function of u and of 

 the s unaccented variable to be determined. But after the introduction 



* This function in its original form is given by 



-. /• p' dx dy dz 



J {{X - xy + (/ - yf + (.' - 2)2}"-^' 



where dx dy dz represents the volume of any element of the attracting body of which p' 

 is the density and x , y , z are the rectangular co-ordinates ; x, y, z being the co-ordinates 

 of the attracted point p. But when we introduce the auxiliary variable u which is to be 

 made equal to zero in the final result, 



jr _ r p dx dy dz 



J{(^a:'-xf-\.{y-yy + {z-zf + u^yr' 

 ■ - .YOii 

 both integrals being supposed to extend over the whole volume of the attracting body. 



