204 Mr. D. D. Heath on Secular Local 



face, X, Y, and Z being the accelerating forces on any particle, 

 we have 



Xdsc+Ydy + Zdz{=dp)=iQ, 



or (which is equivalent) that f (Xcfo? + Ydy + Zdz) must be inva- 

 riable for every point on the surface. Taking the condition in 

 the latter form, we may, after ascertaining the terms of the in- 

 tegral due to each of the forces, leave out all terms which are in 

 themselves independent of the position of the particle, and equate 

 the sum of the variable terms to zero. 



Let, then, a be the radius of a sphere of equal volume with 

 the earth ; p the density of the earth, that of water being unity ; 

 /3 the thickness of a shell of water equivalent in mass to the 

 shell of ice ; and let yS be the cosine of the north polar distance 

 of any elementary prism of ice, co' its longitude, and s (a con- 

 stant quantity) its distance from the centre of figure of the earth; 

 and lei //,, co, and /* be the corresponding data for a particle at 

 the earth's surface ; and let / be the distance between these two 

 points, and let dm represent the mass of the prism. 



Then, as is well known, the part of the integral 



§{Xdx + Ydy + Zdz) 



due to the action of dm is — dm\ -£ = -y • And -% may be ex- 

 panded into a converging series, 



i{r 0+ p i r + p^ +&c .|, 



where V n is a known function of /jl, co, fjJ, and co 1 , of "which it 

 is enough here to say that P n = Q'„Q» + terms of the form 

 K cos k (co — ft)'), the coefficient K being independent of the lon- 

 gitudes, Q n being a well-known function of /jl, 



/ 1 ^ 2 -l)* \ 



\2 w xl-2...rc dfju n )' 

 and Q! n being the same function of yu/*. 



The variable part of -w- is therefore 



*fr{p 1+ P t j + P.J + .fa.}; . . . (i) 



* The successive values of Q 2 , Q 3 , &c. may be derived from the two first 

 (Q =l, Q^/x) by the simple formula (w-f- J)Q»+i = (2»+l)/uQ n — nQ n -i, 

 (Bertrand, Calcul Differ entiel, p. 358). In actual calculation from this 

 formula, it is useful to observe that, having at a previous step determined 

 (n— \)Q n -i, and thence deduced Q w _i, we have only to add these two 

 numbers together to obtain nQ n - \. 



