346 The Rev. Brice Bronwin on the Theory of the Tides. 



force of tlie water be able to produce any sensible effect. For 

 this purpose let 



I)= Vi'^ — 2rt^ cos X + ^'^1 = \/ 2 — 2 cos X = 2 sin ^, /•=;•'= 1, 



r and i^ being the radii of the disturbed and disturbing par- 

 ticles, and X t'he angle contained between them, and neglecting 

 the product of the ellipticity by the height of the tide. Then 

 D is the distance of the two particles, and for this force 



where 



f/M=ysin6W^OT', 



y being the height of the tide where the disturbing particle is 

 situated, 6' the colatitude of this particle, and ■cj'— •srits longi- 

 tude from the meridian of the disturbed particle. Hence 



rP iJ sin ^^d^'dxs' 



2 sin § 



the integral being taken from fl' = o to $' = «-, and from ot' = o 



to ro-' = 27r. Let 



sinW^' 



2sin§ 

 2 



then 



»lso 



cos;)(^= cos 9 cos 6'+ sin 5 sin 0' cos (ot'— ot). 



Where the sea rises above the equilibrium height, i/ is 

 positive; where it sinks below it, y is negative ; and where it 

 is at that height, or where there is no sea, it is nothing. 



Let 



y=h-\-h^&\n^v + h^s\n wcos r -f ^3 sin 2« + &c. 



be the part of y which is constant, or which contains only 

 equations of long periods ; then 



7/ = h'-\-h\sm^v + hc.i 



ancl for this part of y, 



Y=/h'dF'-hsm^v/h\dV>+ &c. 



The accent denotes that 6 is changed into $', and the arbi- 

 trary constants in y into those ofy at the place of the disturb- 

 ing particle, these arbitraries varying from place to place, and 



