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NOTE ON THE NEWCOMB OPEEATOES USED IN THE 
DEVELOPMENT OF THE PEETUEBATIVE FUNCTION. 
By E. T. A. Innes. 
(Eeceived February 27, 1913.) 
(Eead April 16, 1913.) 
In my paper on the algebraical development of the elliptic perturbative 
function (Transactions, vol. ii., part 3, pp. 301-317, 1911) certain recur- 
rence formulae were given by which the Newcomb operators involved 
could be built up from those of a lower order. As therein remarked, 
Newcomb and Chessin had already done the same, but the advantage 
in my method was the brevity of the calculations and the disclosure of 
the recurrence-law which permitted the indefinite extension for primary 
and secondary terms. I now add two tables which will include all the 
most important tertiary terms (namely, all those under the 6th order of 
the excentricities). 
Herr G. v. Zeipel has recently published a paper on these operators 
(Arkiv for Mathematik, Astronomi och Fysik, Stockholm, Band 8, No. 19, 
1912), which gives a very simple analytical solution of the problem. He 
has found — 
4gn3 = ( - 2D + 4:i)UlZl + ( - D + t)Ulzl 
u) (i+i) (i + 2) 
and expressions of the same nature for nj+' and n|+'*. 
If exphcit developments in terms of D and i are required, these are 
the most convenient expressions yet derived, but in practical use they 
suffer from a disadvantage which the older recurrence formulae avoid. 
Herr v. Zeipel's formulae only build up the operators, and it is therefore 
necessary to introduce the i's as has been done, whilst the others build up 
both operator and its object. These operators, when numerical applica- 
tions are made, must be attached to the a'A^ of the perturbative function. 
