The Rev. Brice Bronwin on the Theory of the Tides. 267 



second member of the preceding only by the small quantity 

 Ej^p^' sin^ V cos^ v sin^ ^j. We cannot affirm that this is a very 

 near approximation ; and without more knowledge of the 

 comparative values of Dj and Ej, we cannot express /Sj as we 

 have done /Sj. Dividing the third of (12.) by the fourth, we 

 have 



^ Q A. P , T^ osin vcosu ,, _ . 



C0t^, = C0t^l+E,/j3— -j_^; . . . (16.) 



whence it appears that /Sj may in some cases vary very cotisi" 

 derably. 



Let (o) be the angle which the orbit of the planet makes 

 with the equator, (z) the longitude measured on it from their 

 intersection. Then (o) is the obliquity and {z) the true lon- 

 gitude for the sun, and they are nearly the same for the moon. 

 Also {z) is the hypothenuse, {v) the perpendicular, and (o) 

 the angle at the base of a right-angled spherical triangle ; and 

 (4/) is the base for the sun, and nearly so for the moon. Hence 

 by spherical trigonometry, 



tan 4/= cos o tan z, 

 and therefore 



rf\(/ dz d'h cos^vl; 



~ cos — 5- » -7- = cos o 



cos-'^J/ cos^z dz cos^z 



But 



cos 5^= cost? cos tf/j 

 consequently 



^4/ COSO . -ox 



3- = — 5— = cos 0(1 + sm'' v)s 

 dz cos-* V ^ ' 



neglecting sin'^ v and higher powers. Also 



sin v=- sin osmz; 

 therefore 



-r- = COS 0(1+ sin^o sin^z)= cos of 1 + -sin^o— -sin^ocos2z j 



, 1 • 2 ^ 



= 1 — - sm'' cos 2z. 



2 



neglecting sin^o, &c. 

 But 



vt + e = z—2es\x\{z — Tt)\ 

 consequently 



vdt = dz '- ^ed^ cos (^r — tt ), 

 neglecting e^, &c. 



It may be doubtful whether we should express the value ot 

 ^ in terms of the mean longitude or of the true ; I have chosen 



