ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 399 



the sign / serving to indicate * integrations relative to the variables 



x^, x-i, X3', x/, and similar to the double and triple ones employed 



in the solution of geometrical and mechanical problems. Then it is 

 easy to perceive that the function V will satisfy the partial differen- 

 tial equation 



t/vr ^, d^ ^ n-s dV 

 " ~ dx,^ "^ dxi "*" ^ dx^ '^ du^^ u du ^^' 



seeing that in consequence of the denominator of the expression (1), 

 every one of its elements satisfies for V to the equation (2). 



To give an example of the manner in w^hich the multiple integral 

 is to be taken, we may conceive it to comprise all the real values 



both positive and negative of the variables ar/, x^, x,, which satisfy 



the condition 



the symbol / , as is the case also in what follows, not excluding equality. 



2. In order to avoid the difficulties usually attendant on integra- 

 tions like those of the formula (1), it will here be convenient to notice 

 two or three very simple properties of the function F". 



In the first place, then, it is clear that the denominator of the 

 formula (1) may always be expanded in an ascending series of the 



entire powers of the increments of the variables x^, x^, x„ u, and 



their various products by means of Taylor's Theorem, unless we have 

 simultaneously 



and therefore V may always be expanded in a series of like form, 

 unless the s + 1 equations immediately preceding are all satisfied for 

 one at least of the elements of V. It is thus evident that the func- 

 tion V possesses the property in question, except only when the two 

 conditions 



