^}.(12.) 



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



or 



5^= Fg cos 2/82 cos 2(p + F2 sin 2j32 sin 2<p + F, cos /Sj cos f 



+ Fi sin ^1 sin f. 



This value of {y) (the height of the tide), compared with the 

 preceding, gives 



F2Cos2/32=D2Cos2g2 + E^P^c^s^'^j F2 sin 2/82 -=D2sin2§2 

 Fj cosjSjzsDiCosSj + Ejp^sinvcost;, Fisin/3j = Disin§j. 



From these equations we must determine F2» Fj, /Sg, and /3^. 



In small seas it is very easy to see that the direct effect of 

 the force is very small compared with the horizontal velocities, 

 and therefore that we ought to have D2 much larger than Eg. 

 We shall proceed on this supposition. Therefore if /32 = §2 + '^> 

 A is a very small quantity, and we shall have 



cos 2/82 = cos 2^2— 2 A sin 2^2' sin 2/32= sin 2g2 + 2Acos 2^2 



nearly by Taylor's theorem. Multiply the first of (12.) by 

 the first of these, and the second by the second of these, 

 member by member, and add the products ; and we find 



F2=D2 + E2p3cos2tJcos2g2; . . . (13.) 

 neglecting the small term 



— 2E2Ap3 cos^v sin 2^25 



which containsthe product of the two small quantities Egand A. 

 Dividing the second of (12.) by (13.), member by member, 

 there results 



F 

 sin 2/32= s>" 2^2— rf p^cos^ v sin 2^2 cos 2^3 

 1^2 

 very nearly. If we put for sin 2/83 in this its value before 



given, we find 



A = — ^-j^ p^ cos'^ V sin 2^2» 

 and consequently 



^2 = ^2- ^^P'cos^^ sin 2^2. . . . (H.) 



We cannot positively say that Dj is large compared with 

 E, ; but if we add together the squares of the third and fourth 

 of (12.), we have 



Fi2 = Di2 + 2D,Ejp'^ sin ?; cos v cos g, + EfpSin^ v cos^v. 



Suppose 



Fi = Di + Eip^ sin i;cosr;cos §1. . . . (15.) 



The square of the second member of this will differ from the 



