MR. LUBBOCK’S RESEARCHES 
340 
If the sun or primary be a spheroid, a the angle which the plane of the sun’s 
equator makes with the plane of the orbit of the planet ; and if the longitude 
be reckoned from the line of intersection of the sun’s equator with the orbit 
of the planet ; R is increased by the quantity c 3 &m - w x — - 1 j. } c being a 
constant dependent upon the figure of the sun ; but the partial differential co- 
efficients of this quantity, which are introduced into the values of d e, d «r, &c. 
do not change the form of the expressions for those quantities. 
If the planet move in a medium which resists according to any power n of 
the velocity, if c be a constant and v the velocity, the term 2 cfv 71 + 1 d t must 
be added to 2/dR, 
{o+VaHr'4;} to GS), 
d A' . /d R\ 
C V 
cv 
n — 1 
« - 1 r*'2 
r 4 dT to 
and cv n 
i 
r — — to 
d t 
in the equations of p. 330. 
If the orbit of the planet be supposed to coincide with the plane x y, so that 
s = 0, then by the equations of p. 337 after reductions 
tl a = — 2 c a 2 -f 1 + ecostq 
\ a / 1 — e cos u J 
71 + 1 ,, s 
~ 2 ~ (1 — ecos u) 
do 
n — 1 
n — 1 
de = - / ; — (Inf) 
d=r = - 2 c(^ VIE 2 
\ a / 1 — e cos v J n 
cos o d o 
sin v d o 
71—1 
71—1 
i „ / /x \ o f 1 + e cos v ~) - 77 — sin v C 1 — e 3 cos v ~) . 
de-dsr = 2c - 2 < t- 2 > 2 — - i >d u 
\ a / [1 — e cos v j n (. e J 
The form of these equations differs from that which obtained before, now 
the variations of e, -a and e are periodical, while that of a has a term which 
(J £ 
varies with the time, contains only odd powers of cos o and for that reason 
has no constant term. The periods of the periodic inequalities of all the elliptic 
constants due to the action of the resisting medium are fractional parts of the 
periodic time of the planet. 
