The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 47 



sin </> . . ^, tan d> •, . .x cos (x — S) _,, . , , 



■ . , sm (V, — 0) = ^, and cos (v, — v) — . Ihereiore, by sub- 



smi ^ ' ^ Vdm \ I J cos ' -^ 



stitution, 



sin A, = tan cos (v — S-) ( -: — . ; ) = tan - tan cos (v — b) = 



' ^ \sm I tan ij 2 



sin i / r,\ 

 -. tan cos {v — 3). 



2 cos- 2 



But sin A, = A, — ^A], and sin i cos (u — S) = y -r-. Consequently, ^A — 



J 3 _ tan r (ia _ y' o- da _ »'^ c? \/(l — o^) 



^ ' ~ Z T^ h dt~ ^, ,i :/(T^r^) It ~~ ~ ~i dt ' 



2 cos^ n 2/t cos- - ^ ^ 2ft cos- - 



and A, = r ^- '- + ^lA . 



2AC0S''- 



Now A, = r-r, -3- + = A-3- + e=:A+a; and, therefore, 



„ i (ft 



A= • \u -° + ^^^ (') 



2Acos- 



neglecting smaller quantities, observing that a is of the order of the disturbing 



z 

 force multiplied by sin- - . 



We want now to find --^- and -r— . To do this, we shall transpose ^A', 



and, after taking the partial differential, divide by 1 — |A^, or multiply by 



1 . 1 A' J 1 c '^''^ ^ • / UN cos i da , (/-CT 



1 + -i A- ; and by means oi -rr-r = -5 cos i cos ( w — S-) = - — ■. -^ , and -, — ,- = 

 •' dult r^ ^ ^ sm i dt dbdt 



— a, we shall find 

 r" 



dA r' da a r ./2 - <r\ n ., , ,. 



2Acos^- ^ -^ -" 



