1896.] Earth's Free Eulerian Precession. 189 



9. The effect of the transference of masses of water by ordinary 

 ocean currents and other causes has been considered long ago by 

 Lord Kelvin and Prof. G. H. Darwin, and more recently by Prof. 

 Newcomb, with the result that such disturbances are amply 

 sufficient to originate displacements of the Earth's axis comparable 

 with the amplitudes of the observed changes of latitude. 



10. If the Earth were absolutely rigid but covered throughout 

 by surface waters, the momental difference C — A' which gives 

 the free precessional velocity would thus be less than the one 

 C — A which gives the forced astronomical precession, by that of 

 a spheroid of water of the ellipticity 6 : (=- s \ J ) 1 which is due to 

 centrifugal force alone. This latter spheroid, of semi-axes a(l — fej) 

 and a(l +^ei), has the moments of inertia 



C\ = \<#E, (1 + § ei ), A x = \a?E, (1 - £ 6l ) I 



so that for it C\ — A l = ^a"E l e i , where E x is the mass of a sphere 

 of water of the dimensions of the Earth. 



11. The potential of the attraction of the actual Earth at 

 distant points is, by Laplace's formula, 



Jr (E A + B + C-3I \ 



where E is the Earth's mass, a its radius, and yE/a- = g. In the 

 case of symmetry when A = B, we have thus 



V E G-A /9 • 1N 



- = 5-j- (3 cos 2 - 1) + . . . 



7 r zr 3 



6 being the co-latitude. 



The potential of the centrifugal force of rotation is 



V x — \arr- sin 2 6. 

 Over the surface of the ocean of ellipticity e, given by 



r = a(l + esin 2 0), 



the total potential V+ Fj must be constant : thus 



E G — A 



— (1 — e sin 2 0) s — - (3 cos 2 — 1 ) 4- ^&> 2 a 2 7 sin 2 = const., 



«. 2« :i 



fi * E 3C-A . . . . 



so that - e - g — — - \ co-a^y = 0, 



which gives C — A = %a?E (e — | m) ; 



where m = Qi 2 a/g, is the ratio of centrifugal force to gravity at the 



1 Thomson and Tait, § 821. 

 VOL. IX. PT. III. 14 



