290 Transactions of the Boyal Society of South Africa. 
If we adopt the variations in e and e^ and i and found by other 
investigators, we can draw some further inference from the equation (A). 
Hill ("Collected Works," iv., p. 127) has discussed the maximum and 
minimum values of e and e^, and he points out that the maximum eccen- 
tricity of one orbit corresponds with the minimum eccentricity of the 
other, and vice versa. From Hill's figures we get — 
LC0 (min.) = 9-9991924-10 LC0, (max.) = 9-9999705-10 
LC.^ (max.) = 9-9998597-10 (min.) = 9-9984324-10 
If we write the first equation of (A) thus — 
and substitute the above values of and G(pj, we get — 
St = 0-4044 Sti. Min. of variable factor. 
= 0-4051 St,. Present value (1850). 
= 0-4054 St,. Max. value. 
These equations show that t and t, increase or decrease together, and 
that with considerable precision we have at all times — 
At = 0-405At,. 
It must be remembered that At and At, are the variations of the 
inclinations to the invariable plane of the system. 
Stockwell (" Smithsonian Contributions to Knowledge," xviii., 1870) 
gives the following figures : — 
Maximum value of t + 1, = 1° 29' 35" 
Minimum value of t + ti = 1 1 39 
to which we add the present value =1 15 20-9 (1850) 
Value 2,000 years hence (Le Verrier) = 1 15 1 (3850) 
These values show that with great accuracy we can write the second 
equation of (A) in the approximate form — 
I (7 = 0-405 
C<p + qG^^ = c 1.403 
or making e and e^ variable — 
CAC + (/6iA6'i=0. 
* The argument is complete without these, and logically they should not be used ; but 
their use simplifies the reasoning. 
