1896.] Earth's Free Eulerian Precession. 191 



exceed the other from this cause by more than 7 per cent., even if 

 the Earth were rigid. 



Actually, the diminution of precessional moment due to the 

 surface waters will be much reduced by the elastic yielding of the 

 core, for when the centrifugal force is removed not only the 

 surface but also the bed of the ocean will diminish in ellipticity, 

 so that the differential effect will be smaller. It follows therefore 

 that 6 to 8 per cent, is an outside limit to the actual increase of 

 period of the precession that can be due to mobility of the surface 

 waters. 



12. On the equilibrium-theory here advanced, the portion of 

 the ellipticity of the surface waters which arises from centrifugal 

 force alone, shifts in company with the axis of rotation, thus 

 causing a tide of the same period as the free precession, as was 

 first pointed out by Lord Kelvin 1 . The equation of this ellipticity 

 is r = a(l + e 1 sin 2 #), where e 1 = ^\ T . Thus a shift 86 in the axis 

 of rotation involves a change of level dr/dd . B0 ; which is at 

 the rate of 02 sin 20 of a foot for each second of arc. The 

 reductions of tidal observations in Holland by Bakhuysen, and on 

 the Pacific and Atlantic coasts of North America by Christie, 

 made with this object in view, actually gives a tidal component, 

 of the period of the Eulerian precession, with amplitudes respec- 

 tively about '03, "04, and '05 of a foot, thus corresponding to a 

 precessional radius of the order of a quarter of a second of arc, 

 which is the observed order of magnitude. 



The somewhat close agreement which has been found between 

 the phase of this tide and the phase of the precession tends 

 towards verifying both that the equilibrium theory here worked 

 out is sufficient and that the free precession of period 427 days is 

 fairly constant in amplitude. The amount of the tide as here 

 computed will however be diminished by the centrifugal yielding 

 of the solid Earth ; though, this diminution being regular, we 

 should still find that if the precession were quite regular in ampli- 

 tude, the tide would at each station be proportional to the 

 amplitude of the precession multiplied by the sine of twice the 

 latitude. But it would appear from the observed latitude varia- 

 tions that the amplitude of the precession is altered so irregularly 

 that the harmonic analysis of a long series of observations can 

 hardly bring out anything more definite than a sort of mean 

 amplitude of tide. The actual tide thus produced follows, on the 

 equilibrium theory, the movement of the Pole exactly ; but it 

 can only be disentangled from the observations of aggregate tide 

 by an assumption that it is of a periodic harmonic character. 



1 Collected Paper*, Vol. in. p. 332. 



