ATTRACTIONS OF ELLIPSOIDS OF VARIABLE DENSITIES. 423 



where f 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 « = in (37) we have 



, _ a^ y" z' ^ 



a" + h' "*■ b'-' + li' '^ c" + h' ^ h' ^^ ^' 



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

 h must constantly remain equal to zero, and the equation (38) will always 



be satisfied by the indeterminate positive quantity — . 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, %, as will satisfy the equation (38), of 

 which the last term now evidently vanishes in consequence of the 

 numerator o'. Thus the forms of the quantities A, B, C, D and F" 

 all remain unchanged, and the discontinuity in each of them falls upon 

 the quantity k. 



To compare the value of A here found with that obtained by the 

 ordinary methods, we shall simply have to make n' = 2 in the expression 



(36), recollecting that r(l) = 1, and r (-] =i\/7r. In this way 



, .,,,,/- hdh ^ ,,, , r da 



A = — Aiiraoc X \ -rrr- = — 4nrab c x / -7^— 

 J^ctbc J„drbc 



= + ^a'h'c'x f 4?- = 4-«'*'c' J . , f " ^ 



But the last quantity may easily be put under the form of a definite 



integral, by writing - in the place of a under the sign of integration, 



and again inverting the limits. Thus there wiU result 

 J 47r«'J'c' /•! v"dv 



^ = 'n^~ J 



a •'o 



a + ^«^)(i + ^-/-^^) 



a' ' a- 



