in Laplace' s Theory of the Tides, 233 



powers of x, and) such that -7-, -r-g, -j-r 6 , . . . &c. are all finite 



for every finite value of x. Hence -777, being equal to \/(l — x 2 ) ,-, 



is zero when x = \. 



9. Thus it appears that Laplace's process simply determines 



K 4 to fulfil the condition that -7^=0 at the equator. And the 



assumed form of solution (3) has the requisite convergency to 

 zero when x = 0, for the poles. Laplace's result is therefore the 

 solution of the determinate problem of finding the tidal motion 

 in an ocean covering the whole earth continuously from pole to 

 pole. Whatever other motion the sea could have in virtue of any 

 initial disturbance cannot, except for certain critical depths, have 

 the same period as that of the assumed tide -generating force. 



10. If the sea be precisely of such a depth that some one of 

 the possible free vibrations in which the height of the surface at 

 any instant is expressible by the formula v cos 2ijr, where y]r 

 denotes longitude, and v some function of the latitude having the 

 same value for equal north and south latitudes, has its period 

 equal to that of the tide-generating influence, it is easily seen 

 that the solution of the differential equation (2) gives an infi- 

 nitely great value to a. It is only when the depth has one of 

 these critical values that the arbitrary solutions introduced by 

 Airy and adopted by Ferrel are applicable to an ocean covering 

 the whole earth continuously. 



11. Yet Laplace himself fell into the same error of imagining 

 that the general integration of the differential equation (2), with 

 the proper arbitrary constants, includes oscillations depending 

 on the primitive state of the sea, as the following passage (Liv. iv. 

 chap. i. art. 4) shows : — 



" L'integration de l'equation (4)* dans le cas general ou n n'est 

 " pas nul, et ou la mer a une profondeur variable, surpasse les 

 " forces de i'analyse ; mais pour determiner les oscillations de 

 "l'ocean, il n'est pas necessaire de l'integrer generalement ; il 

 " sufBt d'y satisfaire ; car il est clair que la partie des oscillations 

 " qui depend de l'eiat primitif de la mer, a du bientot disparaitre 

 "par les resistances de tout genre que les eaux de la mer eprou- 

 " vent dans leurs mouvemens ; en sorte que sans Paction du soleil 

 " et de la lune, la mer serait depuis longtemps parvenue a un etat 

 u permanent d'equilibre : Taction de ces deux astres Fen ecarte 

 " sans cesse, et il nous suffit de connaitre les oscillations qui en 

 " dependent." 



* A general equation, of which equation (2) of our numbering above is a 

 pa.ticular case. 



