1878.] 



with the Tides of a Viscous Spheroid. 



197 



non-rigid attachment of the permanent eqnatorial protuberance to the 

 mean sphere. It is shown that the precessional and mitational couples 

 will give rise to certain tides of the second order (varying as the tide- 

 generating force multiplied by the precessional constant), but not to 

 any secular shifting of the surface over the interior, as has been sup- 

 posed would be the case by some writers. 



II. Distribution of Hedt generated by Internal Friction,, and the secular 

 < ' cooling of the Spheroid. 



In the paper on " Precession" it was shown by the theory of energy, 

 that a very large amount of heat might have been generated inside 

 the earth by friction, but the investigation gave no indication as to 

 its distribution. The problem is here considered by finding the 

 amount of work done per unit of time on each element of the interior 

 in the course of the tidal distortion. 



The aggregate work done on the whole globe is found to be the 

 same as that given by simple considerations of energy. The rate of 

 work is equal to the tidal frictional couple multiplied by the relative 

 angular velocity of the moon and earth ; but this simple law arises 

 out of a complex law of internal distribution. By far the larger 

 part of the work done, or heat generated, is found to be in the central 

 portion. 



My first impression was that the large amount of heat> which might 

 be generated, would serve to explain in part the observed increase of 

 underground temperature ; but the solution of a certain problem con- 

 cerning the cooling of an infinite slab of rock 8,000 miles thick, in 

 which heat is being generated according to a certain law of distribu- 

 tion, shows that the frictional heat could not possibly explain a rate 

 of increase of underground temperature near the earth's surface of 

 more than 1° F. in 2,600 feet. 



It follows, therefore, that Sir W. Thomson's investigation of the 

 secular cooling of the earth cannot be sensibly affected by this cause. 



III. The Effects of Inertia in the Forced Oscillations of Viscous, Fluid, 



and Elastic Spheres. 



In the theory of tides used hitherto the effects of inertia have 

 been neglected. It was, however, shown that this defect in the theory 

 could not have an important influence, unless the frequency of the 

 tides was much greater than that of those generated by the moon at 

 the present time. Nevertheless it was desirable to determine what 

 the effect of inertia actually is. 



This part of the present paper contains a second approximation to 

 the theory of tides of a viscous spheroid. 



VOL. XXVIII. P 



