The Rev. Brice Bronwin on the Theory of the Tides. 195 



and -^, —T- are far too small to render it necessary to take 

 account of them, it is obvious that 



since (p=w/ + a constant and very small variable quantities. 

 Therefore 



8Ac«=Sd 



< — n«AM— 2«sindcosd -^ + Tcos2(f — l2)+llsin2(<p — ^2) > 



+ 8ct-{ — M^ sin^ flAu + 2« sin fl cos 9 -^ + "^ sin 2(f — ^.J >. 

 That this may be a complete variation, we must have 



^4'-yMM-2nsinflcosfi^+Tcos2(<p-g2)+nsin2(9-^2)T 



= J-4^~«2sin2^A^_|. 2n sinScosd ^ +"^ sin2((p-g2) I . 

 But if we attend to the value of <p, we see that 



dv: ~ n dt ^ d'srdt ~ n dt'^ 

 Also for the quantities we are seeking we may put 

 dAv_, d^Au _jd^u 



~dr-^^'^' 'Mi -^~w 



d 



since the coefficients only are affected by the operation ■^. 



The preceding equation therefore, when reduced, may be 

 written, 



^1^+^.^ + ^s^^-^^.^^Kf-^A . (8.) 



+ K5COS2((p — g2)=0' 



Leaving out the term — -(r*, if we put the other terms of 

 (7.) in (1.), the result may be written 



dAu dAv Awcosfl _, ^, p \ . a • ^/ p \ ^ 

 "dT "^ "^ "*■ ^i^^ + rcos2(^-g2) + Asm2(^-g2)=0. 



And this may be put under the more simple form, 



LiAM+i^+rcos2(f-g2) + Asin2(^-g2)=0 . (9.) 



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