56 
MR. LUBBOCK’S RESEARCHES 
Let P be the place of the planet, P' its projection on the fixed plane LNP' 
(fig - . 1 & 2.) SN the line of nodes, SL the line from which longitudes are 
reckoned. The angle L S P' = X'. Let S A be the line of apsides, (fig. 2.) 
In the notation of M. de Pontecoulant, vol. i. p. 316, the angle ASN 
— S’ = h M. de Pontecoulant has given expressions for the variations of 
the constants a, g, e 1 , l, / and v in terms of the partial differences of the quan- 
tity R with regard to these quantities. It is easy from these to find similar 
expressions for the variations or differentials with regard to the time of the 
constants a , ®r, e, s, t and v. 
Let SAB be a plane cutting the plane of the orbit at right angles, so that 
the angle SAB = 90°, ANB = (, BSN = ^-v 
— + h ~ _ ll? + Jt — o 
d t- r 2 cos < 2 r a 
r = a { 1 — e' cos (y — a) } 
When r is a maximum or minimum ^ = 0, 
a h 2 
,a cos i ~ 
— 2 a r -f r 2 = 0, whence r 
= a + \/ a — 
h°- 
p. COS i- 
t = a ( 1 + e') 
h ~ =o(l- e' 2 ) 
COS l- 
By the equation of p. 336, line 12, (Phil. Trans. 1830.) 
h — = a ( 1 — e” + e 2 sin 2 1 sin 2 (y — w) ) 
,i- \ / 
p,COSI 
e' 2 = e 2 { 1 — sin 2 1 sin 2 (v — m ) } * = e 2 cos 2 A S B 
Considering R first as a function of the quantities a, g, e', l, i and v, and 
then of the quantities «, e, s, / and v, we have 
= (Sfh + (SI) <•+ G!) “ + Gf) *• + (it) 11 • + Gf) “ 
* The equation I gave, Phil. Trans, for 1830, p. 336, line 17, is not correct. 
