OF ELLIPSOIDS OF VARIABLE DENSITIES. 191 



employed in the solution of geometrical and mechanical problems. 

 Then it is easy to perceive that the function V will satisfy the 

 partial differential equation 



. n-sdV 



~~^ + + ^ " 



seeing that in consequence of the denominator of the expression 

 (1), every one of its elements satisfies for Fto the equation (2). 



To give an example of the manner in which the multiple 

 integral is to be taken, we may conceive it to comprise all 

 the real values both positive and negative of the variables 

 oj/ajj', ... #/, 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 

 integrations like those of the formula (1), it will here be con- 

 venient to notice two or three very simple properties of the 

 function V. 



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^ a; 2 ,...aj 8 , 

 u, and their various products by means of Taylor's Theorem, 

 unless we have simultaneously 



#! = #/, # 2 = tf 2 ', ...... X 8 = x 8 ' and w = 0; 



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 F. It is thus evident 

 that the function V possesses the property in question, except 

 only when the two conditions 



^2 T- 2 ~2 2 



?* + ?* + ?-' + - + ? i<1 and u=0 ............ 



cfcj a 2 u 8 u 8 



are satisfied simultaneously, considering as we shall in what 





