1878.] 



with the Tides of a Viscous Spheroid. 



195 



In forming the theory of tides, it was assumed that the action 

 of the tidal protuberance on any element of the surface of the mean 

 sphere was entirely normal to the sphere, and consisted of the weight 

 of the prismatic element of the tidal protuberance, which stands on 

 the element of surface. This is not rigorously correct, because, if it 

 were so, there would be no couples tending to alter the diurnal 

 rotation and obliquity of the earth. The effects of these couples were 

 considered in the paper on " Precession," but the tidal protuberance 

 was there assumed to be instantaneously rigidly connected with the 

 mean sphere. The present problem is concerned with the non-rigid 

 attachment of the protuberance to the sphere. 



A sphere is supposed to be distorted into any form differing in- 

 finitesimally from the true sphere, and to be acted on by any external 

 disturbing potential. It is then found what tangential stress must be 

 supposed to act across the base of any prismatic element of the pro- 

 tuberance, in order that the equilibrium of that element may be main- 

 tained, the pressures transmitted by the four contiguous elements 

 being taken into account. It appears that if the protuberance has 

 the equilibrium form, due to the external disturbing potential, then 

 there is no tangential stress between the true sphere and the pro- 

 tuberance. But since the tides of a viscous spheroid lag, the form of 

 the viscous tidal protuberance is not one of equilibrium, and there is 

 such a tangential stress across the base of each element of the pro- 

 tuberance. It is obvious that these tangential stresses may produce a 

 continued distortion of the spheroid. 



The problem, as applicable to the earth, is treated in the simple case 

 where the obliquity to the ecliptic is zero, and where there is only one 

 disturbing body or moon. 



The sum of the moments of the tangential stresses about the axis of 

 rotation gives the tidal frictional couple, and its form is found to 

 agree with that found by a different method in the paper on " Pre- 

 cession." 



When the earth's rotation is taken into account, it appears that the 

 component along the meridian of tangential stress at any point of the 

 surface is periodic in time ; whilst one part of the component perpen- 

 dicular to the meridian is periodic, and the other non-periodic. The 

 periodic parts of the component tangential stresses give rise to small 

 tides of the second order (varying as the square of the tide-generating 

 force), and are neglected, but the non-periodic part gives rise to a 

 secular distortion. 



Since the earth's rotation as a whole is retarded, therefore the dis- 

 torting tangential stresses all over the surface constitute a non- 

 equilibrating system of forces, and in order to find the distortion of 

 the globe, they must be deemed to be equilibrated by the effective 

 forces due to the inertia of the slackening diurnal rotation. These 



