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XXIV. On the Dynamical Theory of Deep-sea Tides, and the 

 Effect of Tidal Friction. By D. D. Heath, M.A., F.G.S* 



THE effect of tidal friction on the earth's rotation has lately- 

 attracted some of that general attention which is always 

 paid to any speculation which, however remotely, bears on per- 

 manent and indefinite alterations in the condition of our system, 

 and so appears to bring science into connexion wdth history or 

 prophecy. 



But the study of this effect requires a knowledge of the dyna- 

 mical theory of the tides. And this, so far as I know, is only 

 to be found in Airy's treatise in the Encyclopedia Metropolitans 

 On betaking myself thither, I found that the main principles and 

 typical cases, as regards deep and continuous seas, admitted of 

 being separated from the mass of more complicated calculations 

 in which they lie imbedded; and, moreover, I thought I could 

 exhibit them so as to shed a clearer light on the physical laws 

 which they prove. And I thought that what has given me so 

 much pleasure and insight might not be unacceptable to some 

 of the readers of this Magazine, more especially when I found 

 the Astronomer Royal himself implying that his work is not so 

 well known as it certainly ought to be to eminent mathema- 

 ticians f. 



This is the'history of the earlier and larger part of this paper; 

 the latter part, both what relates to Laplace's theorems, and to 

 the effect of friction, perhaps carry the theory a little onward. 



Kitlands, Dorking, 

 January 12, 1867. 



1. The moon attracts every particle of the earth with a force 

 which is smaller the greater the distance, but which is every- 

 where much less than that with which the mass of the earth acts. 

 The effect is, that the earth remains a coherent, nearly spherical 

 body, having a motion very nearly the same as if it were reduced 

 to a single particle at the centre; but the difference between 

 the effective force producing this motion, and that actually im- 

 pressed at each point, causes a pressure which, combining with 

 the resultant of the internal attractions, modifies the shape of all 

 fluid or flexible parts of the surface. 



What is known as the Bernoullian, or statical, theory of the 

 tides consists in a demonstration that, if the moon were always 

 vertical over the same point of the surface, this shape would be 

 that of a prolate spheroid pointing towards the moon, — with the 



* Communicated by the Author. 



t Monthly Notices of the Royal Astronomical Society, April 13, 1866. 



