[ 343 ] 



XLII. On the Theory of the Tides. 



By the Rev. BiiicE Bronwin. 



[Continued from p. 197.] 



IF, in order to find the terms depending on sin (<p — S^) and 

 cos((p — ^j), we combine u^ and v^ with u^ and Wj, there 

 will result terms with constant coefficients containing the factor 

 fliflTj- And though they might in many places be larger than 

 those before found containing a^j, they would be very much 

 less than those of the first order ; and, moreover, we may 

 suppose them included in those terms of the first order which 

 have constant coefficients. Besides, if we take account of 

 terms thus formed, we ought also to take account of the terms 

 depending on sin 3(<p — ^3) and cos 3{f—SQ), which must be ftt 

 least as large. 



Combining «, and Vi with Uq and Vq^ we find from (3.) the 

 terms 



85(sin2 $ - cos2 $)2n'^Da- cos (f - ^i) + ^^{ (sin^ $ - cos^ $Y2?i^Ca- 



— 2«2sin 9 cos QDcr} sin ((p — ^1) — 83(C + 2 sin 9 cos 5D) 



w2^cos(^-&i)-89(sin^dD + 2sindcos9C)n2^sin(^-g,). 



This, to abridge, may be written 



aS^GV-H'^) cos (f-^i) + S=^K'<^sin (^-^i) 



-89L'|sin(^-g,). 



(15.) 



Observing that -^ =0, the equation of continuity gives the 



terms 





dr 



dQ 



sin(f~gi)-^<r2, 



which may be written 

 -icr2 + (M'cr + N'^)cos(f-g,HC^sin(^-&0. (16.) 



As before, we neglect the small quantity — - <r% since it is 



not of the form of the terms we are seeking ; and we see that 

 (15.) and (16.) are exactly similar to (5.) and (7.), and there- 

 fore by exactly the same steps as those employed upon them 



