of Laplace's Theory of Tides. 547 



moon, it will follow, since \] = u + u' f Y = v + v l , W =«; + «/, 

 that 



is an exact differential. 



The general conclusion from this argument is that, for the 

 calculation of tidal motion in any case whatever, udx + vdy + wdz 

 must be assumed to be an exact differential, the calculation 

 being restricted to terms of the first order, and u, v, w being 

 the components, parallel to axes fixed relative to the earth, of 

 the tidal motion at xyz, as it might at any time be actually ob- 

 served and measured. 



In the Mecanique Celeste, liv. i. chap. viii. art. 34, Laplace 

 asserts that " in the theory of the flux and reflux of the sea we 

 cannot suppose udx + vdy + wdz to be an exact differential, 

 because it is not such in the very simple case in which the sea 

 has no other movement than that of rotation which is common 

 to it with the earth*" It is true that if the values of u, v, 10 at 

 any point include resolved parts of the velocity at that point due 

 to the motion of rotation which the ocean partakes of in common 

 with the earth, the expression udx + vdy + wdz cannot be an exact 

 differential. But this circumstance affects in no manner the 

 foregoing argument, by which I have shown, after fully taking 

 account of centrifugal force, and abstracting from the motion 

 of rotation, that if u, v, w be resolved parts exclusively of tidal 

 motion, udx -\- vdy -\- wdz will be an exact differential. Such 

 values of u, v, w are in fact just those which a theory of tides is 

 required to find. 



Consequently, if [d<f>) be put for udx + vdy -f- wdz, so that the 

 equation of constancy of mass becomes 



p± ds> d^ 



d X * + dy* + ds* ' ( > 



this equation is perfectly general, as respects the problem of 

 tides. By making use of it, a solution of that problem much 

 simpler than that proposed by Laplace would probably be ob- 

 tained. I cannot forbear expressing the opinion that the differ- 

 ence between the views recently expressed in this Journal by 

 two eminent mathematicians relative to certain points of Laplace's 

 theory, may be traceable to the needless complexity of the mathe- 

 matics of that theory, and might be expected to disappear if the 

 simplification indicated above were adopted. 



In vol. xxxix. of the Philosophical Magazine (April 1870, 

 p. 260) I have given a solution of the problem of the tides of an 

 unbounded ocean of uniform small depth on the supposition 

 that the moon revolves about the earth in the plane of the equator 

 at the mean distance with the mean angular motion, and I have 



2N 2 



