88 The Rev. Brice Bron win's Differential Equations 



= sin 2 ~ sin2(z; - S) (i + |- sin 2 <J>) = sin 2 -|Yl + -j-sin 2 A 



sin 2 (w — S) — — sin 2 ~- sin 2 i sin 4 (u— S) = sin 2 -^- ( 1 + sin 2 -^- ) 



1 . i 



sin 2 (v — d) — sin 4 — sin 4 (t> — •&), neglecting quantities 



that are insensible. Make ty = v + ev, = .&+ev, ei> being 

 the regression of the node, and 



M = sin 2 4-(l +sin 2 -M sin 2(4/- 0)--^sin 4 4-sin4(4/-0). 



Let M be the value of M when i and are changed into i 

 and O , these quantities denoting the constant parts of i and 6. 



And let A M = I ( ■ — rfH — =*■ rf $) » where 4> is regarded 



as constant both in the differentiation and integration. This 

 will be the part of M depending on the variable parts of i and 

 $ ; to which must be added the correction at the origin, or 

 i 



J- 



Sill" — 



2 / -rf$ = O. Then the reduction = M n +AM n + 0. 



<s cos i 



We must put for di, dQ, and d$ their known values. Such 

 equations in A M -f O as have like equations in v should be 

 put to the value of v; the remainder, if any, with those of M 

 must be separately tabulated. 



The coordinates of the disturbing body will be easily ex- 

 pressed in terms of v. Thus 



n t + s = v + Ej sin (v — w) + E 2 sin 2 (v — ir) + . 



Or if — = m. 



n 



«' t + m s — m v + m Ej sin (v — it) +; 

 n' t + e' = m v + ta + m E t sin (v — n) + , 

 where w = e' — m i. By this value of »' t + r we can express 



7 J , or 7/' = — , and v', the elliptic rad. vect. and long, of the 



disturbing body in terms of v. 



We must not make me = s', or w = 0, as has always been 

 done in the lunar theory, unless we afterwards expel m v by 

 restoring v', when -sr will vanish with it. 



But I should prefer expressing u' and v' in terms of n' t + e'; 



or rather I would retain v', and express v! = -,-, by means of 



v\ and integrate in the manner M. Hansen has done in his 

 Perturbation of Comets. Thus in an abridged form we have 



