332 
MR. LUBBOCK’S RESEARCHES 
d 2 * = (“)’dr'do 
d s ?' / d.r's\ 2 a v /d B\ /dJ?\l 
j~*+ Uair) - pT^ t + r +7{V^^ + r Ad7') -Hd7)j =°- 
If the orbit of the planet m coincide with the plane x y, s is of the order of 
the disturbing force, of which therefore neglecting the square, r s = r, X x = K 
^- a + r+ ‘S{ 2 f AR + r (^)} =( , 
When the disturbing force is neglected, the integral of this equation is 
r = a { 1 — e cos (u — a)}, v being the eccentric anomaly. 
n t + s — ts — v — a — e sin (u — a) 
X — OT 
tan 
= {mV 
tan 
If Q be put for the quantity ^ 2 J * d R + r ( jy) j* and the constant a, 
which may afterwards be replaced, be omitted for the present, 
r = a {1 — e cos v} — sin vj'a cos v d v + cos vf a sin v d v 
d t 
=A!{ a { 1 — e cos v } — sin „/q cos v d v + cos of Q sin u d y j- d. 
nt + g — -is = v — e sin v — ^ {/« d v — cos f 2 cos v d v — sin vfQ. sin y d y | 
If v = f (n t + £ — «) in the elliptic theory, then neglecting the square of 
the disturbing force, 
= f (n t + g — -a) + 
d.f(?i£ + e — ot) 
d t 
;{/« d v — cos vf 3 cos v d v — sin vf Qsiny j dy 
If £ v, h r denote the values of those parts of r and v which are due to the 
first power of the disturbing force, 
d v — cos vj a cos v d v — sin vf Q sin y d y J 
or = a e sin v S v — sin v f Q cos udu + cos vf Q sin v d v 
= z / Q {e — cos u} d v — , / V sin v d v. 
1 — e cos uJ 1 ’ 1 — e cos v J *■ 
In the elliptic theory, 
ii d t cos o — e 
d o = 
1 — e cos v 
1 — c cos u 
= COS X, 
sin u 
(1 _ e *)i 
± L — sin a 
1 — e cos v 
