58 
MR. LUBBOCK’S RESEARCHES 
Expression 
for the reci- 
procal of the 
radius vector, 
when a < a,. 
f 71 
£ + «!_ 6 )l ecos( „( +6 [6] 
(re — »,) (re + re,) [ 2 re 2 /u, 12 a,- 2 a, 2 ’ 2 a, 3 ’ 4 a, 2 ’"J J ‘ L J 
7i- r 3 (2 w — ra,) 2 ^ _ m, f 4n / a 2 _ a- ^ _ a 3 , 3a 2 ^ 1 
(3ra— re,) (re — re,) i 2 re 2 1 |u, [ (2 re — re,) {. 2 a, 2 2 a, 2 3,0 2a, 3 3,1 4a, 2 3,2 / 
+ - 2^; S « + - J$ j w} | ecos (2» ( _ nf + 2 S - - ») [8] 
[3(3ra — 2re,) 2 m, [ 6 re f a 2 , a 3 , . 3 a 2 , 1 
J ~ r - j (3- - 2 ' »,) { ~ 4^*»- ~ + J 
(4 re — 2 re,) (2 re — 2 re,) y 2n- 
5_ a 3 , J 
2 a, 3 3 ’ 2 2 a, 2 
f* 
-^• 1+ 4 “5^3,2 ~ 4" ~1^ 3 - 3 f f e cos (3 « f 2 re,f + 3 s — 2 s, — w) [9] 
M 2 
[ 3 (4 re — 3 re,) 2 ^ _ m, [ 8 re / a 2 t a 3 A , 3 a 2 t ) 
f (5re -3w,) (3re -3re,)i 2re 2 3 ft\(4n-3»,)l 4a, 23 ’ 2 2 a, 3 3)3 + 4^ 2 3 ’ 4 J 
~i^T 2 * 3 ' 2+ 2 S* 3 ’ 3 “ 4 jecos(4ref -3re,f + 4£-3£,-re7) [10] 
, J 3 (5 re — 4re,) 2 _ ret, [ 10 re / _ a 2 » a 3 , ,3a 2 1 
(6 re — 4 ;j,) (4 re — 4 re,) j 2 re 2 4 ju, [ (5 re — 4re,) l 4a, 2 3,3 2 a, 3 3,4 "“"d a, 2 3,5 J 
— ^ 3 . 3 + y ^^,4 — ^^,5j|ecos(5ref-4re/+ 5s -4s, — w) [ll] 
re 2 I 3 (re — 
2 re, (2 re — 2re,)i 2 
-2w,) 2 ^ _ m, J 2 re f 3 a- A a 3 A a 2 , ) 
re 2 7-2 ~/x t( w — 2 re,) l4a7 2 3,1 ~ 2a^ 3,2 ~ 4a^ 3,3 J 
- ^&3,2 + 2^ _ 2^.3 j | ecos (« t - 2n,« + £ - 2 s, + TO-) 
[ 12 ] 
re 2 
3 (2re — 3 re,) 2 ^, m l \ 
[ 4 re j 
f 3 a ~ jj a ' ^ 1,5 j 1 
(re — 3 re,) (3 re — 3 re,) 1 
2 re 2 7-3 jm, j 
[ (2 re — 3 re,) 1 
14 a, 2 3,2 2 a, 3 3 ' 3 4 a/ 3 ' 4 / 
9 a 2 
4 a, 2 
+ 1 — <Ae — 2— 3 ^3,3 + ■*—&,* U cos (2 re t — 3 re,i + 2 s — 3 s, + rer) 
4 a, 2 *•’] 
[ 13 ] 
re 2 | 
[3 (3 re — 4 re,) 2 m, j 
r 6w j 
^ 3fl2 & __ a ’ £ fl2 b \ 
(2 re — 4 re,) (4 re — 4 re,) "j 
2 re 2 1 f* | 
[ (3 re — 4 7i,) 1 
l4a, 2 3,3 2 a, 3 3,4 4 a, 2 3>i J 
+ 3 “1^3,3— 3 |^3,4 + “i^3,i| ec os (3 re < — 4 re,< -J- 3 s — 4 s, + ot) 
[ 14 ] 
