(19) 
ON THE DEYELOPMENT OF THE PEETUEBATIYE FUNCTION 
IN THE THEOEY OF PLANETAEY MOTION. 
By E. T. a. Innes. 
(Eead April 19, 1916.) 
In the Transactions, vol. ii, part 3, pp. 301-317, 1911, I have a paper 
upon the Newcomb operators used in the algebraical development of the 
elliptic perturbative function. The present paper deals v^ith a further 
extension of the uses of these Newcomb operators. At best the develop- 
ment of the perturbative function is a very cumbersome business, and 
reference must be made to Newcomb's original paper in vol. v. of the 
Astronomical Papers of the American Ephemeris for a full statement. On 
p. 15 Newcomb quotes his remarkable theorem by which the operators, 
acting on the development in powers of e, combine with the operators 
acting on the development in powers of e^, e and e^ being the eccentricities 
of the inner and outer planets respectively. The operators connected with 
e he writes as n", those with e^ as n[J| y', and he proves that 
n;;}; = n»x n„,. 
These operators consist of a string of differential coefficients of the Lap- 
lacian functions hi or a'A,-. 
Thus we have — 
8nJ;^ = [8i3 + (4i2 - 20D + (- 2i - - D'^]5., 
in which — 
D — a—- . 
da 
Operators which are factored by powers of a- — sine squared half mutual 
k 
inclination of orbits are indicated by the affix k thus : fj"-"'. 
Unfortunately Newcomb only shows how the operators can be combined 
symbolically, and therefore in numerical work it is necessary to consider 
the eccentricities of both orbits simultaneously. 
