188 The Rev. Brice Bronwin on the Them-yofthe Tides, 



sary equations, retaining his symbols, which I suppose it will 

 not be necessary to explain. 



From book 4, chap. 1. sect. 4, we have 



aW=^Um^v- icos^wVl+ScosSd) 



SL 



+ -^ sin d cos9 sinucost;cos(w^ + CT— \l/) 



SL 



+ T-qSin^dcos^jycos 2(w^ + «r— vp). . . . (1.) 



This is the part of aV which depends on the action of the 

 sun or moon. That which depends on the disturbing force 

 of the water, arising from the deviation of its form from that 

 of the equilibrium state, is insensible in small seas. And in 

 these the deviation of the attraction of the land from the regu- 

 lar law of gravity will mostly be as great, or even greater, 

 than that of the water ; and its tendency will be in a consi- 

 derable measure to annihilate the eifects of the latter. We 

 shall therefore neglect this force. 



From book 1 . chap. 8. sect. 36, we have, making the den- 

 sity unity and restoring p, which the author had made nothing, 



+ r28z!r(sin2d ^ +2wsin9cosfl^) = -g8j/-8/?+8V. 



In this formula Laplace has very properly neglected the 

 vertical displacement (s), which is quite insensible in compa- 

 rison of the horizontal displacements [u) and (u). He has 

 also left out the terms multiplied by 8;-, which quantity ought 

 to be considered as of the same order with (s). If in this we 

 put r= 1, and make 



8«,=89( -^ —2n sm Q cos ^ 37 ) +^«^ 



(sin2d^+2«sinflcosfl^), (2.) 



the above becomes 



g8y + 8/)=8V-8co. 



This, integrated, gives gi/+p=c + Y—oo, where (c) is an arbi- 

 trary constant. It may, however, contain equations of long 

 periods in (/), but cannot contain the angle w^ + c7— vj/. Now 



