134 On a Problem in relation to the Figure of the Earth. 



Laplace's functions {¥, G, H . . . being functions of //. and a>, and 

 M p M 2 . . . being Laplace's functions themselves), the several 

 terms ought to be identical with the terms of the same order in 

 u or 1+m 1 4-w 2 + ... But, from the manner in which i enters 

 into the second side, it is clear that every term in the series into 

 which the second side is expanded will involve i. But i does 

 not enter at all into the terms 1, u v w 2 . . . 



Hence we must so choose our arbitrary quantities as to pre- 

 vent this contradictory result. The only method of doing this 

 is to make M x = 0, M 2 =0, . . . , also to make F, G, H . . . con- 

 stants, and 



Hence w=l, and r=a; that is, the only surface which suits the 

 conditions is the sphere ; and the law of density such that it is a 

 function solely of the distance from the centre, since F, G, H . . . 

 are independent of fju and w. 



5. This property of the sphere was of course known before, 

 because a uniform spherical shell attracts an external point as if 

 condensed into its own centre. But the present investigation 

 shows that there is no other kind of surface which possesses the 

 property. Hence no changes in the arrangement of the mate- 

 rials of a body can be made so as to preserve the external attrac- 

 tion unaltered, except uniform and complete spherical concen- 

 tration and dispersion of matter to or from one or more fixed 

 centres in the body. 



6. I trust I have now vindicated the truth of the two important 

 propositions which have been noticed recently in your Magazine, 

 while I have at the same time taken the opportunity of correcting 

 and improving what I had already written*. Both these propo- 

 sitions — viz. (1) that if the earth's form be a spheroid of equili- 

 brium the arrangement of the earth's mass without doubt ac- 

 cords with the fluid law, and (2) that Bessel's method of ap- 

 plying the theory of least squares admits of improvement so as 

 to take account of local attraction at the reference-stations (which 

 it has hitherto altogether overlooked) — bear upon the fluid theory 

 of the earth. For it is only by geodetic means, which the second 

 proposition seeks to improve, that it can be ascertained whether 

 the mean figure of the earth is a spheroid of equilibrium or not, 

 and therefore whether the first proposition represents the state 

 of the earth's mass. 



In the ocean, which covers so large a portion of the globe, we 



* See Phil. Mag. for June and July 1866. 



