234 
MR. LUBBOCK’S RESEARCHES 
The preceding- results are obtained by the direct integration of the differen- 
tial equations : I shall now show that they coincide with the results obtained 
by the variation of the elliptic constants. 
The equations for determining the variations of the elliptic constants are, 
d a = — 2 a- n d t 
d s 
d R 
d R 
ii= - -(I - V\ -e‘)^d( + 2o>»^'di 
{e = a«y\-_£ —— .dK d 
p v / 'Cl£ p n 77T 
d ct 
, an \/ 1 — e- d R , , 
dar= — — 1 d t 
e d e 
, an d R , . 
d v = — ,, -r- d t 
sin i V 1 — ■ e 2 d » 
d t = — 
a n 
d R 
sin i V"l — e- d v 
See the Theor. Anal. vol. 1. p. 330, or The Mechanism of the Heavens, p. 231. 
In these works R is used with a contrary sign to its acceptation in the 
M<3canique Celeste, which I have followed. 
When the square of the eccentricity is neglected in the value of the radius 
vector, the equations may be employed in the following shape : 
, n o d R , , , , an d R , , , 0 „ d R , . , an e d R , A , an d R , ^ 
dg ede da 2dg e dzu 
Jf ndt d £ = n d -Rd t 
— = — ^ 1 1 + e cos (n t + g — •us) | + cos (n t + g — ■&) $ e — e sin (n t 4- g — nr) (5 g — $ ot) 
+ 2 e cos (2 n t + 2 g — 2 nx) $ e — 2 e 2 sin (2 n t + 2 g — 2 ot) (8 g — 8 cr) — e sin (n t + g — vs) 8 g 
b j i cos i (nt — n t t) 
(n - n,) a , 
n f _ a 2 j _ a 3 ? , 3 a 2 , "1 
{i (n — w,) + n J. 1 4 a,' 2 3 ’* — 1 2 a, 3 3, ‘ 4 a, 2 3 >‘+ 1 J 
| cos {nt — zj) cos (i (n t — n t t) + n t — 
+ sin {nt — zu) sin (i (nt — n t t) + n t) — | 
