OF ELLIPSOIDS OF VARIABLE DENSITIES. 215 



After what precedes, it seems needless to enter into an ex- 

 amination of the values of V belonging to other values of the 

 density p, since it must be clear that the general method is 

 equally applicable when 



where /is the characteristic of any rational and entire function. 



The quantity A before determined when we make u = 0, 

 serves to express the attraction in the direction of the co-ordinate 

 x of an ellipsoid on any point p, situated at will either within or 

 without it. But by making u = in (37) we have 



x* if z 2 (? 



1 = + + * + * .......... (38) ' 



and it is thence easy to perceive that when p is within the 

 ellipsoid, h must constantly remain equal to zero, and the equa- 

 tion (38) will always be satisfied by the indeterminate positive 



2 



quantity -5 . When on the contrary p is exterior to it, h can 



no longer remain equal to zero, but must be such a function of 

 x, y, z, as will satisfy the equation (38), of which the last term 

 now evidently vanishes in consequence of the numerator o 2 . 

 Thus the forms of the quantities A, B, C, D and V all remain 

 unchanged, and the discontinuity in each of them falls upon 

 the quantity h. 



To compare the value of A here found with that obtained 

 by the ordinary methods, we shall simply have to make ri=2 in 



= i\V. 

 In this way 



, 7 , , f hdk ,-,, , [ da 



A = -4:7rabcxl -^- = - 4?ra b c x I - w - 



J^ a be J^ a be 



da ,,, , da 



