206 Mr. D. D. Heath on Secular Local 



due to the action of the ice-cap is 



^^{X,% + A B % r - + &c.\ (A) 



To find the part due to the attraction of the earth, we may 

 conceive the solid nucleus as made up of the same volume of 

 matter of density 1 (i. e, of water), and of another equal volume 

 of the actual density diminished by 1. The former part unites 

 with the sea to make up a homogeneous spheroid. Now, r being 

 measured from the centre of gravity of this homogeneous sphe- 

 roid, and a being the radius of a sphere of equal volume, we 

 may assume r = «(l +?/)=«{ 1 +a 2 Q 2 H-a 3 Q 3 + &c.} ; and then, 

 neglecting y 2 , the attraction of this spheroid is known* to yield 

 towards our integral 



3r 



+ w{^ + ^ + &c. + ^ + &c.}; 



or, putting -(1— y) for --, and leaving out the constant term, 

 it is 



r?^{*Qi+«A+*e-} +«{?&4!A + &.}. (B) 



The rest of the mass of the earth will act as if collected at its 

 centre of gravity. And if this were at any sensible distance 8 

 from the centre of figure, we should have for this action (precisely 



as for dm, only expanding in powers of - instead of k, so as to 



converge) an expression which, 8 being along the polar axis, 



becomes 



mass f ., ~ 8 ^ S 2 1 

 — |l + Q,-+Q v +&c.j; 



and when we come to put all the variable parts of our integral 

 together and equate corresponding terms to zero, there will be 

 no other term (while we confine ourselves to the first order of 

 magnitude) involving Q,, except this one multiplied by 8. We 

 must therefore have 8=0; or the nucleus must remain central, 



as before the disturbance. And then writing - (1 — y) for -, 



* I use both Laplace's formulae here without giving the proof. His own 

 is notoriously defective ; and I do not remember to have seen the proof of 

 the second formula (for the attraction at the surface) correctly given in 

 print. It follows, however, when the function Q n has been independently 

 ascertained, from Laplace's mode of reasoning. 



