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



viscosity proportional to, or a function of, rate of alteration of 

 strain would (unless its coefficient were excessively large) be 

 ineffective in preventing the assumption of the configuration of 

 fluid equilibrium in the slow motions here considered, and free 

 precession of periodic character would therefore be non-existent. 



8. The influence of the mobility of the surface waters on 

 the period of the free precession has however yet to be considered. 

 For them also, if they are sufficiently deep, the time of sub- 

 sidence of a disturbance will be small compared with the period 

 of the precession : so that the principle already employed will 

 still have an application. The simplest case to consider first is 

 that in which the waters cover the whole surface. When the axis 

 of rotation changes, the centrifugal strain will be shifted to the 

 new axis; thus the ellipticity of the underlying solid Earth will 

 be altered owing to its elastic yielding, while that of the ocean 

 surface will be altered by a different amount easily calculated 

 from the conditions of fluid equilibrium. 



If on the other hand the ocean consisted only of a thin layer, 

 its law of depth would be changed considerably, and this may 

 involve sensible currents in it. Taking the angular momentum 

 arising from the velocity of these currents (which will diminish 

 the correction due to the surface waters) to be negligible in the 

 actual case, the precessional couple will be as before that of the 

 configuration the Earth would assume if the centrifugal force were 

 removed ; it will be that of a spheroid of water with the free 

 surface that would exist on the removal of that force, together 

 with that of the underlying solid Earth with its centrifugal force 

 removed, but with its specific gravity reduced by unity. 



If the underlying Earth were absolutely rigid, the effect of 

 the layer of water on the precessional couple would be to diminish 

 it by the couple arising from a spheroid of the same density as 

 water, with ellipticity equal to that part of the ellipticity of the 

 ocean surface which is due to centrifugal force alone : if the 

 underlying Earth is yielding, the ellipticity of this effective water- 

 spheroid must be that of the ocean surface less that of the solid 

 Earth, both due to centrifugal force. In any case, it is clear that 

 under these circumstances the free ocean will cause an increase 

 in the precessional period. This, as above stated, is on the sup- 

 position that the ocean is on the average deep enough for the 

 angular momentum of the currents created in the alteration of 

 its free surface to be neglected: if for instance it were very shallow 

 it could obviously have no appreciable effect, and that would be 

 in part because the momentum of the currents would compensate 

 the effect of altered ellipticity, in part because its surface would 

 not then have time to attain the equilibrium form. 



