270 The Rev. Brice Bronwin on the Theory of the TideSt 



E . . 1 . 



sin /3,= sin Jcy— ^r- sin k^ cos k^ sin v cos v+ -sin^ocoskySmQ.z 



E . 1 . . 



= sin 7^1 — —■ sin ^i cos Jc^ sin u cos ?;+ - sin ocos^^ sini?cos\I/ 



nearly, and therefore 



•p 1 



/3] =k] — j^ sin Jci sin v cos u+ ^ sin o sin v cos \^ 

 nearly. 



We might find in the value of the terms D3Cos3(<p— §3) 



and D4 cos ^((p — §4); the coefficients Dg and D4 being, as 

 before, functions of fl without / or 2 ; but these terms must be 

 very small. 



From (1.) it is easy to see that there will also be terms of 

 the form 



l+m sin^ o sin 2z+p sin^ cos23r + qe sin [z—Trj + se cos (z—tt), 

 the coefficients /, m, &c. being functions of fl; but as they are 

 very small, we shall not attempt to determine them. If, how- 

 ever, we wish to make 8w a complete variation relative to such 

 terms, we must make the coefficient of Sw to vanish, or 



sm'^ 9 — ^ +2« sm 9 cos 9^7 =0, 

 ar at 



which will give an equation of condition. The arbitrary of 

 integration will now be a constant independent of 9. And we 

 shall have 



8«j = a9(— 5 — 2wsm9cos9-^), 



or 



8w = S9r^( -5-2- — 2« sin 9 cos 6^1 —2nrh' sin^ d -j-, 



according as we make dco a complete variation relative to Q 

 only (which, indeed, it is already) or relative to both Q and r. 

 The equation of continuity will become 



• adu . 



sm -jT + cos 9 11=0, 



or 



dirh) Jdu A 



as the case may be. I believe we must complete the variation 

 for both r and 9. 



Gunthwaite Hall, near Barnsley, Yorkshire, 

 September 7, 1849. 



[To be continued.] 



