190 Mr Larmor, On the Period of the [May 25, 



equator, viz. is giu- The value of e, the ellipticity of the sea- 

 level, is about 2V4 1 '- so that the vaiU e of C - A is just one half of 

 that which would belong to the actual free surface in the absence 

 of centrifugal force. 



[We observe, incidentally, that 



V-E . . , 3cos*fl-l . 



dV 

 and g = — -j orr sin 2 # 



= 7 — (1 — 2e sin 2 #) — 7 — (e — \m) (3 cos 2 — 1) - orrt sin 2 



= # {l-(fm-e)sin 2 6>}, 



where g is the value of g at the Pole ; which is Clairaut's formula 

 for gravity. 



Again, the value of (C — A)/C, derived from the precession of 

 the equinoxes, is •00327 2 : thus 



= fa 2 #(e - im)/-00327 = '35a?& 



If the Earth were homogeneous of density 5^, we should have 

 C = 0'4>0a?E : thus the increase of density towards the centre 

 diminishes the axial moment of inertia by 12 per cent. 3 ] 



Thus for the actual Earth, C — A = \Ea? . -^ ; from which the 

 value for the centrifugal water spheroid, that is 



W 



is to be deducted, leaving the effective moment reduced in the 

 ratio of 72 to 82. 



In this case therefore of a rigid Earth covered by a fluid ocean, 

 the period of the free precession would be increased in the ratio 

 of 82 to 72 or 1*14 to unity by the presence of the surface waters. 



In the actual case the surface waters do not cover the whole 

 Earth, so that this is only a superior limit. Moreover the distri- 

 bution of surface waters is not symmetrical, so that the result of 

 deducting the centrifugal spheroid of water will be to leave an 

 effective solid with moments of inertia all unequal, and the path 

 of the instantaneous axis of rotation on the Earth will be an 

 ellipse, not a circle. But one axis of the ellipse cannot possibly 



1 Col. Clarke's Geodesy, p. 305. 



2 As quoted from Leverrier in Thomson and Tait, § 828. 



3 As to how this fits with Laplace's hypothetical law of internal density, cf. 

 G. H. Darwin, in Thomson and Tait, § 828. 



