46 The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 



We suppose here that a and a', and therefore also A and A', are small, as in 

 tlie case of the moon and the old planets. 



Making for a moment r, the projection of the radius vector, and v^ the 

 longitude on tlie fixed plane, and comparing the differentials of the areas des- 

 cribed on this plane and that of the orbit, we have 



cos i rdv — r] do^ = 7" cos" dv,, 



and cos i dv = cos" <p dv^ — cos" (p (dv — dA); or 



(cos- — COS i) -V- =cos'' '^^17 ! *^'" ( ^ ^^^ o "~ ^^^ ) ~2 = ^^^ ~r • 



Whence we derive 



o . « . „, r^ dA 



2 sm'' - = sm'' + -r — ,- cos"^ 0, 



^ il (it 



~>7 - - ' " A. ( 2 sin^ ^ - sin^ 

 at r cos- \ 2 



And putting (P- for sin^ 0, 1 — (T for cos^ 0, 



dA h /„ ■ 2 * 



^— = -—- 2 sin'' ^ — «r 



The equation fZ&= cos w/6 gives t/0= -.dS-, andd (0 — ^) = 



° cos I \cos t 



(5) 



1 U/a ; or. 



2sin- 



d (e-B) 



cos J 



■:— fZS. And by integration 



fsm-- 



— S = 2 J r d.^ = o suppose. 



cos 2 



(6) 



Let A, = (w - 3-) - (v, - 0), where v^ = v - A. Then 



sin A^ = sin (v — ■&) cos {v, — 6)— cos {v — 3-) sin (f, — 6). 

 But by the well-known theorems of spherical trigonometrj', sin (r — 3) = 



