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XXV. On an Alleged Error in Laplace's Theory of the Tides. 

 By Sir William Thomson, F.R.S.* 



1. \ IRY, in his article on Tides and Waves in the Encyclo- 

 ■** pedia Metropolitana, gives a version of Laplace's theory 

 of the tides which has the undoubtedly great merit of being 

 freed from certain inappropriate applications of tC Laplace's co- 

 efficients/' or, as they are now more commonly called, " Sphe- 

 rical harmonics," by which its illustrious author attempted, not 

 quite successfully, to design a method for taking into account the 

 alteration of gravity due to the tidal disturbance of the surface 

 of the sea in the solution of his dynamical equations. 



It is the repetition of this attempt for each of the " three spe- 

 cies of oscillations'" {Mecanique Celeste, Liv. iv. arts. 5., 7., 9.), 

 and the preparation for it [in the course of working out the fun- 

 damental differential equations (art. 3.)] by the assertion that " a, 

 b,- c, a 1 are to be rational functions of ^ and i/l~ [*?," that 

 throws a cloud over nearly the whole chapter (Liv. iv. chap, i.) of 

 the Mecanique Celeste devoted to the dynamical theory of the 

 tides, and fully justifies the following statement with which 

 Airy [Tides and Waves, art. (66)] introduces his own version of 

 Laplace's theory : — 



" It would be useless to offer this theory in the same shape in 

 " which Laplace has given it ; for the part of the Mecanique 

 ''Celeste which contains the Theory of Tides is perhaps on the 

 u whole more obscure than any other part of the same extent 

 " in that work. We shall give the theory in a form equivalent to 

 " Laplace's, and indeed so nearly related to it that a person fami- 

 u liar with the latter will perceive the parallelism of the successive 

 " steps. The results at which we shall arrive are the same as 

 u those of Laplace." 



2. The only good thing lost in Airy's treatise through the 

 omission of the spherical harmonic analysis is Laplace's complete 

 solution for the case of no rotation and equal depth of the sea 

 all round the earth. When the earth's rotation is taken into 

 account, or when the sea is of unequal depth, the differential 

 equation to be solved takes a form altogether unsuited for the 

 introduction of spherical harmonics; and Airy's investigation is 

 substantially the same as Laplace's, except the judicious omission 

 of the unsuccessful attempts referred to above. 



3. In giving Laplace's solution for the semidiurnal tide with 

 the change of gravity due to the change of figure of the water 

 not taken into account, Airy points out what he believed to be 

 an error, so serious that, after correcting it, it was " needless to 



* Communicated bv the Author. 



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