The Rev. Brice Bronwin o« the Theory of the Tides. 339 

 I {2phi-2 ^) Fj cos 2 A + Sp'nF'g }• sin 2(^' -/S'g) 



- (sy n - 2 -^ ) Fg sin 2 A cos2 ((p' - /S'^) = 0, 

 and 



tan 2((p'— j3'2)= tan 2(n^ + CT— rj;'— ^'2) 



( 2y« — 2 -7T- ) Fo sin 2A 



dt ) 



2p'nF'2+ { 2p'«— 2 -^ j Fgcos 2A 



which gives the time of high water. 



If we take account of the two small terms Fj cos((p — /3j), 

 F'j cos (f'— /3'i), we may substitute in their diiFerentials the 

 value of ip' given by the preceding formula, and thus correct 

 it; but the correction would be too complicated for practical 

 purposes. Dividing the numerator and denominator by 



it becomes 

 where 



^/ f „/ V Fg sin 2A 



tan2(^'-/3'2)=p,^^^p-^^^, 



, rfA , JA ^ ndt 

 ^ dt ndt 



very nearly, since yw = «—cv'. 

 We have found 



d-^—dzi 1—7; sin^ cos 2a; j; 



therefore, making d and o the same, since their difference is 

 small, 



d^>^=dz'(l — ^ sin^ o cos 2^ j, 



and 



d{^-'\,')=dz—d^-'\s\\\^ o{dz cos 1z-dz^ cos 2^). 



Also 



vf/^=c?^rri — 2^ cos (^r— 7r)~); 

 whence 



^ =y(l + 2^cos (s-tt)), -^ =v'(l + 2e'cos(5r'-7r')). 



Z2 



