402 NEWTON S TRTNCIPIA. 



two luminaries must then be supposed to travel round the 

 earth in respectively a month and a year. Then the pro 

 blem is to determine what motion will take place in any 

 wide expanse of ocean when acted on by these forces, the 

 earth and sea being supposed to have a rotation round a 

 fixed axis. 



But such is the difficulty of the investigation, that if we 

 wish to get any result at all, we must make some suppo 

 sitions that are not quite applicable to the earth. We 

 must suppose the earth to be entirely covered by water, 

 whose depth is the same along any parallel of latitude, 

 which may be expressed by the formula 



y = l (lq cos 2 9), 



where is the polar distance of the parallel of latitude, / 

 the depth at the equator, q a constant depending on the 

 rate at which the depth decreases as we approach the pole. 

 The investigations of Laplace are very difficult : the same 

 results have been obtained by a different but simpler pro 

 cess by Professor Airy in his article on Tides and Waves 

 in the Encyclopedia Metropolitana. 



The result of Laplace * is that there are &quot; trois especes 

 d oscillations.&quot; As these unite without confounding them 

 selves, they can be considered separately. 



Des oscillations de la premiere espece. These are tides 

 of long period, and depend on the positions of the lumi 

 naries, the depth of the sea as compared with the radius of 

 the earth, the time of rotation of the earth, the force of 

 gravity at its surface, and the latitude of the place, but 

 not on the hour of the day. Formula are found to express 

 their magnitude, and these, when no allowances are made 

 for friction and other causes of resistance, are not the same 

 as those given by the equilibrium theory. Laplace, how 

 ever, supposing the resistance to vary as the velocity, and 

 the tide to be exceedingly small (an assumption founded 

 on observation), shows that we may calculate these oscilla- 



* Mec. Celeste, vol. ii. p. 313. 



