188 ON THE DETERMINATION OF THE ATTRACTIONS 



variable u, depends on the property which a certain function V* 

 then possesses of satisfying a partial differential equation, when- 

 ever the law of the attraction is inversely as any power n of the 

 distance. For by a proper application of this equation we may 

 avoid all the difficulty usually presented by the integrations, 

 and at the same time find the required attractions 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 independent variables, each of which may be 

 marked with an accent, in order to distinguish this first system 

 from another system of s analogous and unaccented variables, to 

 be afterwards noticed, and V 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 unaccented ones also formed exactly on the model of 

 the corresponding one in the value of V belonging to the origi- 

 nal problem. Supposing now the auxiliary variable u is intro- 

 duced, and the s integrations are effected, then will the resulting 

 value of V be a funtion of u and of the s unaccented variables to 



* This function in its original form is given by 



p'dx'dy'dz' 



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

 which f> is the density and x 1 , 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, 



p'dx'dy'dz' 



I, 



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

 body. 



