The Theory of Planetary Motion. 
23 
The next operation indicated by Hoo introduces — 
e\nt^\ + e%?n^:? + e'e\nl% 
and so on, and of these Le Verrier only included the first of each. The 
point is, that with exactly the same labour as Le Verrier used for the lower 
terms, we introduce in addition a crowd of higher terms, some of which may 
be sensible. 
Newcomb, on pp. 27-32, gives the expressions for the operators Ylj and 
riQy/ so far as regards development to the 8th powers of the eccentricities — 
these or the modification of them already given in Transactions, vols, ii, 
part 3, and iii, part 2, 1911 and 1913, must still be used ; but the tables for 
Wyji, which Newcomb gives on pp. 32 to 42 are not required. Similarly for 
k 
the II. The number of pages involved shows pretty closely the ratio of the 
saving — viz., as 41 pp. are to 10 pp. 
Johannesburg, March 27, 1916. 
P./S^.— The process outlined above can be generalised. If we have any 
development involving the distances of two planets from the sun, and 
suppose both eccentricities to be zero, then the effects of the eccentricities 
can be introduced bit by bit so far as they are sensible. Thus, if with 
eccentricities zero, we have a term 
f{a)C{i'c/ - ig + 6) 
then the Newcomb operator P/,/ will introduce the effect of ej e-^f and 
change the angle from — 
i'g' - ig ^ 
to— 
(i' + ncf - - j) g + 0- 
A numerical example of this new view of the use of the Newcomb operators 
will be found in the Astronomical Journal for June 30, 1916. 
Sejjtemher 9, 1916. 
