The Theorij of Planetary Motion. 21 
0 
The question is if we operate on Pj with say the operator no,i = 
1 . . . ^ 
^(2{ + 1 + D)> which reduces 11^,0 to l\j i, etc., so as to obtain Py,i, can we 
. ^ . . . 
we include P±j_2& in this operation? We can see that this is so for 
-t 0 
j positive because the operator Y\j{i) is equal to the operator li if i is 
replaced by i — h. This will also include the negative suffices, because we 
k k 
pass from Uj to U-j by replacing i by — (i — 2^). 
Hence — k k 
P-j -2n( — i — Jc) = Pj + 2n {i — h). 
Similarly for the external planet the operator Dq/CO equal to 
n (^ + 2A;), 
and— k k 
Po, -j'-2n( — i— h) = Po,j' + 2n(i — Jc). 
Hence when we have reduced each term to the form — 
PC& C{i'g' + ig) - PS6 S(ig' + %)* 
PC9' C(i'g' + ig) - PS^'S(i'g' + ig) 
2 2 
PC9"C(i'g + ig) - PS6"S(iy + ig) 
we can operate on the sum so as to include the effect of the eccentricity 
of the outer planet — and vice versa. 
As already mentioned, an absolute check is obtained by considering 
firstly one eccentricity nil and then oj)erating, and then considering the 
other nil in its turn and operating. The resulting sums for each term must 
be identical, and they will be so if all sensible terms are included and the 
calculation is correct. 
Each operation introducing e removes the term from the argument 
ig' -f ig to the next argument i^g^ + (i + ^)g, and the introduction of 
e" removes it 7i times. 
As an example we give a portion of the process as applied to some of 
the terms which go to make up the great inequality of Jupiter and Saturn. 
Suppose, firstly, that Jupiter's orbit is treated as being without 
eccentricity, 
We have then — 
elPo,3 (2)C[5^^' - 2^ + 2{w' - w) +0(w' + w)] 
1 
+ e^a'Po,! {S)C[bg' - 2g ^ S(w' - w) + l{w' + w)] 
+ eicr4Po,-i(4)C[5/ - 2g + - w) -\- 2{w' + w)] 
+ terms of the 9th order. 
* Leaded S indicates " sine." 
