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

 We must now substitute for <p and •^ their vahies fl + w and 



sin <p= sin fl+wcosfl, cos<p= cosO—wsinfl; 



then, leaving out the terms of the first order and retaining 

 those of the second only, we shall find 



8Aco = Sfl4 (sin^e— cos2 6)2wM^— sinScosfl-^ j- 



^ r ^ . . « ^ du . . . / d^v du dv\ "I 



+ 8«rj (cos^fl— sin^ Q)2nu-Tr +2smflcos9( m^ '^dtdi/J 



/sin^d^+Snsindcosd^'l ...... (3.) 



where in the general case 



dQ dvr a9 tt'sr 



but for those terms which do not contain -cr, 



We shall make 



tt2stAcos2 (^-^2), V2=Bsin2(f-§2), «i = Ccos((p-^,), 



i>i=Dsin(9-?i)j 

 also 



these two last containing only terms of long periods, and in 

 the first four <p being put for nt + '^—^. 

 Since r=l, 



/ "■ 289* 

 If also g=: I, then 



'^ "289* 

 We must tberdbre consider n as being a very small quantity, 

 and V, the mean motion of the planet, is much less. Conse- 

 quently we must n^lect such terms as 



d<r dr d^a- d^r da^ df^ 

 ""Tt* ""Tt' !¥' If' dt^' dt^' 

 as they contain the factors m or v^. But though these quan« 



