30 
MR. LUBBOCK’S RESEARCHES 
In the general case, when q, t 2 are the inclinations of the orbits of the planets 
P and P t to any plane, the direction of which is arbitrary, 
cos j ; = cos «, cos < 2 •+• sin^ sin i 2 cos (v { — v 2 ) 
. o i t 1 — cos., cos» 8 — sin sin j 3 cos (v, — v 2 ) 
sin- 2 = 2 
The part of R which is independent of nt, n,t 
{ b cl r g 
- -jf- + 4^5 ( 1 “ cos 1 , cos — sin I, sin < 2 cos (v, — v a ) - — g-L j i Sil 
( 3 a , (a 2 + a,®) , 1 , x ] 
“ \ - 25 T- is.. ) ", cos (® - W< ) | 
1 | I*o • tan I 
“"'“✓l+tan*. - 5 *’ 7TTS^ = ,a "‘ near *y- 
= m, j — a + + ,a " s >« — 2 tan i, tan !,««(», — — e* — e, ! | * a ,i 
[3-1, (a 2 + a, 2 ) , I . .1 
- { 2^ 6 *.o ~ 2a/ ~ b ^j 6 6 ' C0S (OT “ w *> } 
= wi ( j — + g-^5 | (tan i, cos v, — tan i 2 cos v 2 ) 2 + (tan j, sin/, — tan < 2 sin < 2 ) 2 — e 2 — e, 2 j b 3>l 
f 3 a , (a 2 + a ; 2 ) ,1 , J 
- 1 2 a/*o - ^ . } e cos («r - w,) j 
and if tan < sin ? =^, tan i cos p = q, this quantity 
= m i{“ if + 8^“ { (Pl “ P2)2 + ' 9l ~~ 9o)2 ~ e °' “ e ' 2 } 4 ».» 
- \ 2^ 63,0 ^ 63,1 J e e - C0S ~ j 
which evidently agrees with the result given by M. de Pontecoulant, Th6or. 
Anal. vol. i. p. 363. All the other coefficients of terms multiplied by the 
squares and products of the eccentricities are susceptible of reductions similar 
to those in the two preceding pages, and finally; 
= m t < 
[ ~ a , + 2 a, 2 ( Sln 2 
c, 7 
[0] 
+ »»/] 
' « ( rn ..o l L e * + e 7 
)“^f + ^sin^A.o+IM 
~ a* \ COb 2 2 , 
a (e 2 +e, 2 ) 
+ a , 8 ft 3.o 
— 4 i 3i2 ) | cos (n t — n ( t + e — e ( ) 
[1] 
