The Rev. Brice Bronwin on the Theory of the Tides, 265 

 -sin^ 9 4 (-A) = -S^2 sin^Z- 



Hcos^^ j 





= - 4«2 sin2- tan^ - (3 + cos 5). 

 When z=l, 



(cos 5 A — sin Q -^ j sind= sin Q cosflA=«, sin d cos 5. 



Therefore 



T)^= -^^nttc^ sin^ - tan^ « (3 + cos 9), D^ = waj sin 5 cos fl. 



These results have been obtained by substituting the values 

 of A from (8.); and it may be observed that all the functions 

 of (i9) are discontinuous at the equator, and only hold from the 

 pole to the equator. By suitably changing the arbitrages we 

 may now write, 



=D2Cos2(!p — €2) + I^iCos(<p--^i) 



A A' 



Dci—a^ sin^ - tan^ « (^ "^ ^®^ ^)» -^i ~ ^i ^'*" ^ ^^^ ^* 

 If we make 



(10.) 



- =p=H-ecos(a— tt), 



where («) is the mean distance, [e) the eccentricity, (2) the 

 longitude, and (tt) the long, of the apse of the planet ; and if 

 we also make 



Eo= :: — 3 sin^ 9, E. = ^ sin 9 cos 5, 

 we have from (1.), retaining only the same terms as heretofore, 



— = Egfi^ cos^ V cos 2(p 4- Eip^ sin v cos v cos f. 

 For these terms, therefore, 



i/= 



aV 



or by substitution, 



3/=(D2 cos 2^2+^2^^ cos^ v) cos 2<p + Dg sin 2^2 sin 2(p 



+ (Dj cos §1 + Ejp3 sin v cos u) cos f + Dj sin ^1 sin f. 



i/=F2Cos2(9-^2) + FiCos(^~^i); • • (n.) 



Assume 



