162 Wisconsin Academy of Sciences, Arts, and Letters. 



If, then, we wish to ascertain the amount of attraction 

 or repulsion exerted upon some material point whose 

 co-ordinates are f, g, h, by a heterogeneous mass or an 

 electrified surface, the integrals (B) or (C) will enable us 

 to write down the values of the component attractions 

 along the axes X, Y, Z at once. But to evaluate these 

 integrals is always difficult ; in some cases impossible. A 

 careful inspection of them, however, will show that if we 

 separate out from each one of them the quantity 



r r r pdxdjdz 



_ r r r p dx dy dz 



=Y, say, then 



the whole expression for X = -j— 



<( (( « 



« Y = ^\ P) 



^- dF 



Here, then, is a quantity Y, which, when once a value 

 is found for it, will give us the values of the attractions 

 X, Y, Z, by a single differentiation of it with reference to 

 X, y, or z. Tills quantity is the Potential Function which 

 we have before explained ; as is evident on a comparison 

 of the two expressions. 



To find Y is generally far easier than to directly inte- 

 grate the expressions for the components of force. Indeed, 

 Laplace has given a general mode of expanding Y into a 

 series, which is both simple and beautiful. To explain 

 methods of finding the value of Yin different cases, is not 

 the object of this paper ; * but merely to call the more 

 general attention of students of Physical science to the 



* This is very fully done in Laplace (Book III, Chap. I, § 4) ; Green's 

 Matli. Papers, edited by Ferrers— Thompson & Tait's Nat. Phil., Vol. I. 



