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

 If we differentiate (8.) and (9.) relative to t, and put 



we shall have two other equations ; and from the four we 

 easily find the values of 



. dAu dAv 



^"' ^''' St' It' 



which will all be of the form 



P sin 2(<p — §2) + Q <^os 2((p — §2)' 

 These being put in the value of IAm, and 'the integration 

 effected relative to -Br, we shall have 



Aco = P, sin 2(^-^2) + Qi cos 2(^-^2), . . (10.) 



where V^ and Q, are of the order na^a-^ noc^r. 



In order to add these terms to those of the first order, we 

 must change, in page 265 of the second paper, T)c^c,os2{<^ — ^^ 

 into 



D2 cos 2(^-^2) + 1>3 sin 2((p-g2)> 

 and then the first and second equations (12.) in the following 

 page will become 



F2 cos 2^2= ^2 cos 2^2 + Egp^ cos^u— D3 sin 2&2I ^ 1) 

 Fg sin 2/82 =D2 sin 2^2 + 03 cos 2§2. ... J 



But here, it must be remembered, D2 is changed from its 

 former value by the addition of small terms containing o- and 

 T, and Dg is of the order na^<r or na^f as it contains these in 

 all its terms. 



By the process employed in the paper just now referred to, 

 we find 



F2=D2 + Ep3cos«wcos2g2 (12.) 



Also 



and thence 



'^2=^2-^P'cos«n2g2+^. . . (13.) 



But, as before observed, D2 is not exactly the same here as 

 in that paper. In (18.) of the same paper we must add to the 



D 



Talue of jSg the small term Tjiy-j and the quantities Dg, Dg re- 

 quire to be developed. 



The quantities cr and t are those parts of the values of u 

 and V which do not contain f , and may therefore be supposed 



