1878.] 



of Viscous and Semi-Elastic Spheroids. 



421 



which satisfies the condition of expansibility as a series of solid 

 harmonics. 



When the disturbing potential wr l Si is zero, and when r=a + Si is 

 the equation to the free surface initially, then the equation to the 

 surface at the time t is given by r=a-\-a\, where 



(qivai ,\ - 

 — 2 t . * 

 2(^ + l) 2 +l J 



This gives the law of the subsidence of inequalities on the surface 

 of a viscous globe under the influence of simple gravitation ; and it is 

 suggested that some light may possibly be thrown thereby on the laws 

 of geological subsidence and upheaval. It appears from this formula 

 that inequalities of wide extent will subside much more quickly than 

 wrinkles. 



The rate is found at which a rotating spheroid would adjust itself 

 to a new form of equilibrium, when its axis of figure is not coincident 

 with that of rotation ; and the law is established which was assumed 

 in a former paper. f 



The case is next considered where Si is a surface harmonic of the 

 second order, multiplied by a simple time harmonic — that is to say, 

 Sj=:S cos (vt + rf). This is the assumption appropriate for the tidal 

 problem. The forces in this case do not form a rigorously equilibrat- 

 ing system ; but there is a couple of the second order of small 

 quantities called into existence, the consideration of which is deferred 

 to a future paper. 



It is then shown that if wi 2 S cos (vt + rj) be a term in the tide-generat- 



19u/y 



ing potential, and if tan e= — — , the tide of the viscous spheroid is 



2giua 



equal in height to the equilibrium tide of a perfectly fluid spheroid 

 multiplied by cos e, and the tide is retarded by e -5- v. It is next proved 

 that the equilibrium tide of a shallow ocean overlying the nucleus is 

 equal to the like tide on a rigid nucleus multiplied by sin e, and that 



there is an acceleration of the time of high water equal to — -. 



2v v 



This theory is then applied to the lunar semi-diurnal and fortnightly 

 tides, and tables are given from which the following is extracted — 

 the coefficient of viscosity being expressed in gramme-weights, centi- 

 meters, and seconds. 



* I write "exp." for " e to the power of." 



f " On the Influence of Geological Changes on the Earth's Axis of Rotation." 

 Phil. Trans., vol. clxyii, Pt. I, p. 282. I take this opportunity of correcting a slight mis- 

 take in that paper ; the formula in the fourth line from the bottom of p. 301, should 



X) E 1 



run , = . = - — cotqc + &c. The mistake arose in copying out the 



2dphc' i 2epkc 4 qc ■ ° 



formula, and does not affect the subsequent arithmetical results. 



