Algebraical Development of the Elliptic Pertiirbative Function. 307 
The presence of the D operator in no way alters the tables, thus — 
0- (-2i-D)n°-2[i;. 
0 = D'^(-2i-D)ri°-D'^2ri;. 
And so on. 
As pointed out in the tables, the forms for n_'/,:^' or ri":_"' are included. 
It must be remembered that such an operator as cannot act alone ; in 
actual work it will be joined to another of at least the same order but 
with a positive suffice, the final form being n_;;; or \IJ{\, and so on. In fact, 
although of primary form, such are closely connected with the secondary 
terms. As the secondary, tertiary, aud higher terms are seldom required, 
it is, as already mentioned, advisable to make them dependent on the first 
order terms. In the tertiary terms we first meet with the quantity i-, but 
it can be eliminated by the introduction of an ^'D operator which will 
make the numerical work simpler. 
The tables for the secondary terms (Tables II. a and II. b and III. a 
and III.B) require no farther explanation — the examples attached to each 
show their use. 
The paper concludes with a symbolic expression for the non-periodic 
or secular part of the perturbative function, which by simple multiplica- 
tions will give the expansion to any powers of the excentricities and 
mutual inclination. Examples are added, which can be compared with 
the known results given by Newcomb and other investigators. 
The happy introduction by Newcomb of the two operators — one of 
which is purely symbolical — D and n, has vastly simplified the develop- 
ment of the perturbative function. The curious properties of the D 
operator have been dealt with in the Montlily Notices of the Eoyal 
Astronomical Society for June and December, 1909. 
Johannesburg, 
Septefnber 5, 1911. 
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