= L2+ ^ Eg sin^ cos 2 (r|/ + ^2) + SeE^ cos (\I/— 2^2— '»") 



The Rev. Brice Bronwin on the Theory of the lides, 269 

 nearly, these may be replaced by 

 F^= Kg— Ej cos 2^2 sin^ f — E2 sin o sin 2^3 sin sy cos 4/ 

 + 3^E2 cos 2^2 cos {z—ir) + 4^E2 sin 2^2 sin {z—v), 



E 1 



182= G2 + ir^ sin 2A-2 sin^ «'— o Hg sin o sin v cos rp 



— QeJrlci sin (z — t) — 57^ si n 2^2 cos (2 — tt ) . 

 Or, if we please, since 



sin^ 5y=sin^ o sin^ 2= tt sin^ » — ;;r sin^ cos 2^, 

 2 2 



Fg = 1.2 + - E2 sin^ cos 2(« + ^2) + S^Eg cos {ss — 2^:2 — w) 



nearly, 



= L 



nearly, 



E 1 



182= G2+ ^rl-s'" ^^2 s'n^ V— - Hg sin o sin v cos ^—^eU^ 

 2U2 -^ 



sin (^I/— tt) 



nearly by neglecting the last term, which is very small, as also 

 are some of those terms containing this quantity which have 

 been retained. In the above 



L2= Kg— - E2 sin^ o cos 2^2= ^2 + ^3 cos o cos 2^2 



Jd 



nearly. 



Making p= 1, and ^i = ^j, in the small terms, we may make 



Fj = Dj + Ej cos ^, sin u cos t;. 



But this will not be very near the truth unless Dj be some- 

 thing larger than Ej; and we cannot conveniently express /S^. 

 But if Dj be considerably larger than Ej, we have 



sin jSj = sin ?j ( 1 — — cos ^1 sin v cos v \ 

 nearly. Also 



sin §1 = sin ^1+7 sin^ cos k^ sin 23 



nearly, neglecting the term containing e. Hence we easily 

 find 



