relation to the Figure of the Earth. 335 



any one of them. Thus, taking the first, 



cos ^Ir 

 P. is ultimately greater than — h*-> 



u ( + 3 u i + 3 cos^ 



.*. - — o "• ls ultimately greater than -7= — 7 rr^~t 



« + o l \l-\-6y 



and is therefore infinitely great for values of u greater than 1 

 when i becomes infinite. This is what was to be proved. 



3. Having shown that the surface of the imaginary body is a 

 sphere, I have to find the law of its density. The original equa- 

 tion of condition now becomes 



Suppose 



^11 C 2iT C l 

 J-1J0 Jo 



1 pr i+2 dr= 



pr i+2 V i dfidcodr = 0. 



F(a, ft, co) 



= F + F i +...+F i + 

 a series of Laplace's functions. Then 



'2jt 



rx 



'" sw+iJ-J. 



2tt 



41 + 3^+.. +(2* + l)Pi+ ..\h\ i d^dco = 0. 



The first side of this equals 47rF'f by a property of Laplace's 

 functions, Y' t being the same function of /// and co' that F; is of 

 fju and a), 



.-. Wi=Q; hence also Fi = 



for all positive values of i, 



I pF~ rt, ar=FQ. 



[ + 2 dr=Y n . 



This is independent of jjl and co ; and therefore p is independent 

 of /jl and co, and is a function of r only. 



4. I come then to this conclusion, that the imaginary body 

 can be only a sphere, with its density (positive and negative) 

 varying as any function of the distance from the centre of the 

 sphere. 



Hence (as I showed in my paper in your Number for August 

 1866) no changes in the arrangement of the materials of a body 

 can be made so as to preserve the external attraction unaltered, 

 except uniform and complete spherical concentration or disper- 

 sion of matter to or from one or more fixed centres in the body. 



J. H. Pratt. 



Jutog, March 7, 1867. 



