44 
MR. LUBBOCK’S RESEARCHES 
Equations (2 n — 4w ( ) 2 
which serve 
to determine 
n~ 
the coeffi- 
(3 n — 5 w,) 2 
cients of the 
1 1* 
inequalities of 
the reciprocal 
of the radius 
(4 n — 6 w,) 2 
n~ 
vector. 
(5 n — 7 ?q) 2 
n 2 
(« +• «/) 2 ^ 
O '68 
. wi, A 
~ r 64 H a q 6i = 0 
r 6i — r 6i + ^ a q 65 = 0 
1 “ 
I m / A 
r 66 — r 66 + a ?66 — 0 
V' 
*57 — r 67 + — 1 a q 6l = 0 
- r 66 + — i- a y 68 = 0 
f* 
In order to obtain the values of the coefficients of the inequalities from these 
equations when the cubes of the eccentricities are neglected, as has been the 
case throughout, the values of r 0 , r x , r 2 , r 3 , r 4 , r 5 , found from the first six equa- 
tions by neglecting the terms multiplied by e 2 , may be substituted in the suc- 
ceeding equations, which will then serve to determine r 6 , r 8 , r 9 , &c. and these 
values of r 6 , r s , r 9 , kc. being substituted in the terms multiplied by e 2 , of the 
equations which determine r v r 2 , r 3 , r 4 , &c. more accurate values of those quan- 
tities may be obtained. All the other coefficients of which the general sym- 
bol is r with a numerical index at foot, may then be obtained in succession 
without any difficulty. 
^3 = A_ ±r±*dt 
d t r- r*J d X 
Let r denote the elliptic value of r, then 
dx_/i 2/ij 1 _ 1 /*d R j t 
d t r 2 r t r\J d X 
= 1 -f -|-+ A e 4 + 2e^l + cos (n t — u) + 31 ^1 + A e ^cos (2nt — 2^) 
11 101 
+ L- e 3 cos (3n t — 3 st ) + e* cos (4)ii-4s?) 
0 4 
-f — e 1 cos (3 n t — 3 sr) + — e 4 cos (4«i-4w) 
b 3 
I) 
cos (n t — w) e 
0-1) 
cos (2 n t — 2 ra - ) 
