The Rev. Brice Bronwin on the Theoty of the Tides. 189 



if we make/?=0, this will belong to the exterior surface, and 

 we shall have 



I3/=C+V— CO, (3.) 



which gives (t/), the height of the tide. 



From the place last referred to we have also 



d{r^s) 2 fdu dv u cos ^ \ _ ^ 



which is the equation of continuity. In this equation the term 



dir^s) 



-J—- ought to have been left out in consistency with what the 



author had done in the other formula. The quantities (m) and 

 {y) are many hundred, or even some thousand times larger 

 than {s) ; it cannot therefore be allowable to express this 

 exceedingly small quantity in terms of them. Leaving it out, 

 and differentiating relative to (/), there results 



. .( d^u d'^v \ .du ^ ,.. 



sm9( :j7t: + T— 7:)+ cosfi^- =0. . . . (4.) 

 \dddt d-sfdt/ dt ^ 



By means of several hypotheses, very wide of the actual 

 case of nature, Laplace has contrived to integrate this equation, 

 with all its terms, relative to (r), and to make the terms in the 

 result all of the same order of magnitude by the introduction 

 of the depth of the sea into the larger ones as a factor. But 

 such a result as this cannot be admitted with any tolerable 

 regard to accuracy. 



The condition that Sco may be a complete variation is 



d {cPu ^ . . -, dv\ d f . ^ .d^v ^ . . . du\ 



or 



d^u „ . . , d^v d / . „, d%\ ^ . ^ , d^u 



. . . d^v d / . „. d^v\ ^ . A . 

 ~2n sm 9 cos fl ^ — y. = -j7\ sm'' 9 -^75 ) +2n sm cos 9- 

 d'srdi rt9\ dr/ 



dmdi^ d't^di ~ d$\ dt^/ ' dQdt 



which by (4.) may be reduced to 



-. — To =-^\sm2 9-7-^)— 2wsm^9-7-. . . . (5.) 

 d-aidt^ dQ\ dt^J dt ^ ' 



From (4.) and (5.) we must determine u and v. To abridge, 

 make (p = w^ + 'B7— vj/. We shall take account of terms depend- 

 ing on this angle only ; these in their most general form may 

 be represented by A sin /f + B cos /f. But A and B, being 

 functions of 6 and /, may be developed in series, the single 



