Belations hetiueen Jupiter and Saturn. 291 
This simple formula shows, as Hill has already pointed out from other 
considerations, that as e increases or diminishes decreases or increases, 
and that when e is at a maximum is at a minimum, and vice versa. It 
further shows that the variations of the eccentricities preserve the ratio 
— 0-405 x-\ Hill's Tables of Jupiter and Saturn do not give the varia- 
tions of the eccentricities directly, but from them I have found the 
centennial variations to be — 
Jupiter +34-06" 
Saturn -71-63 
These figures give a ratio of —0-409^, which is a close accordance,, 
especially when it is remembered that Hill's variations include the action 
of Uranus and the other planets. 
It is remarkable that the ratio of the mean motion of Jupiter to that 
of Saturn ( = 0-4028) is so close to the value of q ( = 0-4052) and that the 
present ratio of the great inequality in the mean longitude of Saturn to- 
the corresponding inequality of Jupiter (by Hill's tables =0-4113) is again 
not widely different from q. These approximations may be fortuitous. 
Summary. 
The following relations, which are practically rigorous, connect the 
variations of the eccentricities and inclinations to the invariable plane of 
the planets Jupiter and Saturn : — 
Jupiter. Saturn. 
Ae= -0-405^'Aei. 
e ' 
Ae= +0-405At,. 
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