lO* The Rev. Brice Bronwin on the Theory of the Tides. 



very complicated ; but it would be useless to find them as 

 functions of 5, since they contain arbitrary constants which 

 can only be determined by observation; we may therefore 

 write the quantity last found thus, 



l(Ga--H^^cos2(f-g2) + 8crK(rsin2(<p-g2) 



-8fiL^sin2(^-g,) 



> (5.) 



where o- and t are the only variable quantities in the coefficients. 



The coefl&cients here will contain the factor w«2, which is 



the same as in the like terms of the first order ; besides this 



they will have the factor cr, or -jr, -tt. They will probably be 



insensible in most places, but in some places they may per- 

 haps be as large as the largest of the variable terms in the 

 coefficients of the terms of the first order. 



In the equation of continuity, the terms of the second order 

 depending on Mj, v^ are easily found to be 



which for the reason before stated may be neglected. 

 Those depending on u^, v^, combined with Uq, % are 



{^^(d<T cosfl\ cos 5 6?(Ao-)'] ^, p\ , r^„dr 



sin2(<p-g2)--<r«. 



To abridge, we shall write this 



-i.r2+(M<r + N|) cos2(^-g,) + 2A|sin2(^~g,). (7.) 



Collecting all the terms of (2.) and (5.), and adopting new 

 symbols to abridge further, we have 



8Ac«=89 

 /^_2wsin9cosS^+Tcos2(^-^2) + Osin2(<p-^2)J^ 



+ 8^|sin29^+2nsinflcosfl^+^sin 2(^-^2) J-. 



But since Am and Aw are of the form 



P sin 2(f - gg) + Q cos 2 (<p - ^g)* 



