268 The Rev. Brice Bronwin on the Theoi-y of the Tides. 

 the latter. By means of the above formulae we have 



d^—cvdt—d^=dz\ c — 1 + ^sin^ocos2^r — 2^Ccos(2r — tt) >. 



As (§) cannot contain a term increasing with z or ty we must 

 have c=l ; then 



d^ 

 and 



=dz< — sin^ o cos 'iz—2e cos [z—ir) V, 



§=^+ — sin^osin 2<3'— 2^sin (2"— tt). . . (17.) 



Change [k) into [k^ and (^j), and we have the values of (§2) 

 and (^i). Whence by Taylor's theorem, 



sin 2§2=sin 2 kct-\- - sin^o cos 2k^ sin 2z—^e cos2^2S'n('2'— "■)» 



cos2^2~cos Ik^— ^sin^ o sin 2/^2 s^" ^^ + ^^ ^'" ^^2 ^'" (2— t) 



nearly. 



Putting for f its value in (13.) and (14.), and 1 — sin^t; for 

 cos^ V, and neglecting the very small terms containing e sin^ v, 

 we have 



F2=D2 + E2 cos 2^2— E2 sin^ «>cos 2^2+ ^^£2 COS2I2 cos(2— tt), 



E E 3^E 



^2 = ^2- grf" ^^" ^^2"^ 20"^^"^ '^ ^^" ^^2- ^^sin2g2Cos(2-9r). 



Substituting for §2 its value from (17.) in these, and neglecting 

 some very small quantities, we find, making to abridge. 



K2=D2+E2C0S2X:2, G2 = ^''2~~o|f ^^'^^^2» 





E 

 H2= 1 — jy cos 2^2» 



F2= Kg— E2 cos 2^2 s'n^ iJ — — E2 sin^ sin 2^2 sin 2^ 



+ S^EjCOs 2^:3 cos (^—tt) + ^^Eg sin 2^2 sin {z—it) 



£ 1 



182=62+ ^jl- sin 2A:2sin^w— - H2sin^osin22— 2^H2sin (z— w) 



'2 

 3eEo 



2D2 



sin 2^2^08(5"— tt) (18.) 



Since 

 sine sin 2«=2sinosin^cos«=2sin i;cos<2r=2 sin i;cos^|/ 



