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 £ — _ ? n V ' 1 -- (1 — V\ — e-) ^dt+ 2a 2 n^d* 
uc — e K ’ d e da 
d e 
_ an \/ 1 — e 2 
(1 - \/l 
' d g e a vs 
d vs = — 
a n 
*/1 — e~ d R 
e d e 
d t 
_ _ an _ 
sin < V 1 — e- d i 
an d R 
sin i V'l — e- dv 
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 
M6canique 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: 
d a 
rt o d R j , 
2 a-n —— d t 
d e 
d £ 
dvs = 
an d R , . , n „ d R , . 
-— d t + 2a 8 n -j— d t 
e d e da 
, ane d R , . , an dR , , 
d e = -j- d t + — t— d * 
2 d £ e d vs 
If 
a 
T 
a 
r 
$ = J* ndt d £ = 3 an d Rd t 
e cos (n t + £ — vs) ^ -f cos (n t + e — vs) $ e — e sin (n t 4- £ — vs) (S e — S nr) 
+ 2 e cos (2 n t + 2 £ — 2 vs) S e — 2 e 2 sin (2 n t + 2 s — 2 vs) (S e — 5 vs) — e sin (n t + s — nr) 8 g 
n a 
b l i cosi (nt — n t t) 
(n — n,) a, 
n J a 2 , _ a 3 , ,_ 3 a 2 , \ 
(n — n t ) + 7 i j l 4a ( 4 3,i—l 2 a t 3 3 ’* 4a ( s 3)8 + 1 J 
| cos (nt — vs) cos (n t — n t t) + n t — vs^ 
+ sin (nt — vs) sin (nt — n t t) + n t) — vs^ | 
