198 Mr. G. H. Darwin on Tides of a Viscous Spheroid. [Dec. 19, 



The first approximation, being that given in the paper on " Tides," is 

 here used to give a value to the terms introduced in the equations of 

 motion by inertia. Physically the terms so introduced are equivalent 

 to an addition to the bodily force which tends to produce the tidal 

 distortion. The problem is then treated by a process parallel to that 

 used by Sir W. Thomson in his statical problem concerning the strain 

 of an elastic sphere. The analytical investigation is long and com- 

 plicated, and it will here suffice to state the result with regard to the 

 form of the tidal protuberance, when the tide -generating potential 

 is of the second order of harmonics. It is as follows : — If a be the 



radius, w the density, g mean gravity, and 0=— , v the "speed" of 



the tide, ?y the alteration of phase ; so that rj -t- v is the " lag," and v 

 the coefficient of viscosity. 



rm, 79v 2 • , 19^V 



Then rj — sm rj cos 17 = arc-tan 



150g hfywa 2 



And the height of tide is equal to the equilibrium tide of a perfectly 



fluid spheroid multiplied by — 



fi . ^v 2 \ 



COS rj I 1 + \ 



This shows that the defect of the first approximation was such that 

 for a given speed, the lag is a little greater, and for a given lag, the 

 height of tide is a little greater than was supposed. 



It is then shown that this correction to the theory of tides will 

 scarcely make any appreciable difference in the results of the integra- 

 tion, by which the secular changes in the configuration of the earth 

 and moon, were found in the paper on " Precession ;" and especially 

 that it makes no difference as to the critical relationship between the 

 month and day, for which the rate of change of obliquity vanishes. 

 The most important influence of the new theory is on the time, and it 

 appears that the time occupied by the changes, above referred to, is 

 overstated by perhaps -^th part. 



A comparison is then made of the preceding theory with that of the 

 forced vibrations of a fluid sphere. This shows that when 77 is zero 

 (i.e., when viscosity graduates into fluidity), the which occurs in 

 the above expressions should properly be \ or y 7 -^. The discrepancy 

 between the 79 and 75 is explained by the fact that in approaching 

 the problem of fluidity from the side of viscosity, we suppose in the 

 first approximation, that the motion of the interior of the sphere is 

 vortical, whereas in reality it is not so. 



In conclusion, it is proved that analysis, of almost identically the 

 same character as that for the problem of the viscous sphere, is appli- 

 cable to the case of an incompressible elastic sphere, and that inertia 

 has the effect of increasing the ellipticity of the tidal spheroid, as given 



