192 ON THE DETERMINATION OF THE ATTRACTIONS 



follows the limits of the multiple integral (1) to be determined 

 by the condition (a)*. 



In like manner it is clear that when 



the expansion of V in powers of u will contain none but the 

 even powers of this variable. 



Again, it is quite evident from the form of the function V 

 that when any one of the -s + 1 independent variables therein 

 contained becomes infinite, this function will vanish of itself. 



3. The three foregoing properties of V combined with the 

 equation (2) will furnish some useful results. In fact, let us 

 consider the quantity 



where the multiple integral comprises all the real values whether 

 positive or negative of o^, &,,...# with all the real and positive 

 values of u which satisfy the condition 



a x , a 2 , . . . a g and h being positive constant quantities ; and such 

 that we may have generally 



a r > a/. 



In this case the multiple integral (5) will have two extreme 

 limits, viz. one in which the conditions 



x z x z X* u* 



-*$ + -** + . + -*i + j- a = l and u = a positive quantity... (7) 



* The necessity of this first property does not explicitly appear in what follows, 

 but it must be understood in order to place the application of the method of inte- 

 gration by parts, in Nos. 3, 4, and 5, beyond the reach of objection. In fact, when 

 V possesses this property, the theorems demonstrated in these Nos. are certainly 

 correct : but they are not necessarily so for every form of the function V, as will 

 be evident from what has been shewn in the third article of my Essay On the 

 Application of Mathematical Analysis to the Theories of Electricity and Magnetism. 

 [See pp. 2327.] 



