344 The Rev. Brice Bronwin on the TJieory of the Tides. 



we shall find equations similar to (8.), (9.) and (10.), and con- 

 sequently that hkU and Au are here of the form 



Fsin(f-gi) + Q'cos(<p-gi), 



P' and Q' having o-, -^, or -jr as factors in every term, and 



also nay P' and Q' are therefore of the order na^^<T, na-^r. 



As the terms previously found are of the same form as A?/ 

 and Ap, the sum of the results will be of the same form and 

 of the same order of magnitude. 



We must now change DjCos (<p — ^i) of vol. xxxv. p. 265 

 into 



Djcos ((p — §i) + D(jsin (<p— §i). 



Then instead of the third and fourth of (12.), we shall have 

 F^ cos/3j = DiCos§i + Eip^sini?cost?— D^sin ^{] .^ . 

 Fj sin /3i = Dj sin ^i + DqCos §1. J 



Whence we have 



Fi2= Di2 + 2DiE,p3 cos g, sin v cos »+ E^y sin^ v cos^ v, 



neglecting 



— 2DoEip^ sin ^1 sin v cos v+Ti^, 

 Suppose 



F, = Di + Eip^cos§i sin wcos?;, . . . (18.) 



as in (15.), page 266, except that D^ has changed its value by 

 the addition oTsome small variable quantities. 



We cannot, however, affirm (18.) to be a near approxima- 

 tion unless we are sure of Dj being considerably larger than 

 Ej, nor can we reduce the following, 



sin /3i = -pJ sin ^i + jp^ cos §i, 



but upon the same supposition. 



We must now change Di into Di+w'sin^w, and put for 

 cos^, its value from (17.), page 266 (paper referred to), or 



cos&i= cos A:j — - sin^o sin ^, sin2^ + 2fsin^isin(2— tt). 



Thus, neglecting some very small quantities, we shall find 

 Fi=D, + Ei cos ^isinvcosv + m'sin^u + S^Ei cos ^isin»cos» 



cos (z— 7r) + 2^E, sin Ati sin ucos w sin (z— tt), . . (19.) 



the accuracy of which will depend upon the supposition of D, 

 being considerably larger than Ej. On this supposition 



E . D 



sin j3i = sin ?j — j^(>^ sin &i cos ^i sin v cos ^ + r^ cos ^j, 



