JRelations between Jupiter and Saturn. 
289 
"We take the following figures from Hill's tables : — 
Le = 8-6835317-10 LC0 = 9-9994938-10 
Le, = 8-7486589-10 LC(^, = 9-9993165-10 
i-i-i.^V 15' 20-9" LC(i + t,) = 9-9998957-10 
and thus have for the value of the constant c — 
Lc = 0-1471664 c - 1-4033512 
= 1-9693950 
We can now make a rough discussion of the last equation, viz. — 
C^(/) + 0-81 Cr/>C0,C(t + tx) + O-16 = 1-969. 
As no cosine can exceed unity, and C^, C^i, and C(i + ti) are essentially 
positive, it is at once evident that at all times the C quantities must each 
be less than but yet exceedingly close to unity. If each is taken as unity 
the equation fails to balance, but only just fails, for we then have — 
1 + 0-810445 + 0-164205 = 1-974650, 
whilst is actually = 1-969395. 
Thus the eccentricities and inclinations of the orbits of these two 
planets can never deviate greatly from their present small values. 
Both t and can be determined in terms of e and e^, and vice versa. 
We have, in fact — 
8^/2 
'(gC^i- 
f Cv>- 
-c) (^001 — C(/)H-c) 
4cC^ 
f C0- 
-c) (0(^ — gC0i + c) 
4c20(/)i 
The first pair of these equations are ill-suited for finding the inclina- 
tions, because gC(/)i + C0 is very nearly equal to c. It is better to find 
t and by trial from the first equation of (A) ; we thus find — 
,. = 54' 4-3" \^^''^- 
