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THE ALGEBRAICAL DEVELOPMENT OF THE ELLIPTIC 
PERTURBATIVE FUNCTION USED IN THE THEORIES 
OF PLANETARY MOTION. 
By R. T. a. Innes, F.R.S.S.Af. 
(Read October 18, 1911.) 
The methods of computing the motions of the planets devised by 
Lagrange and Laplace and used by the great planetary investigators of 
last century such as Le Verrier, Hansen, and Newcomb, require an 
expansion of the reciprocal of the distance between two planets. The 
most general expansion is an algebraical one, which, if it is to be useful in 
practical work, requires that the eccentricities and mutual inclinations of 
adjacent planetary orbits should be small — these conditions are fulfilled 
for all the major planets and for the majority of the minor planets of the 
solar system. Hence, great attention has been devoted in the past to this 
algebraical expansion. The most useful, but at the same time the most 
complicated development, is in terms of the mean anomalies of the planets 
concerned, and it is this development which is dealt with here. References 
may be made to the following works : — 
Le Verrier, Les Amiales de VOhs. de Paris, vols. i. and x. 
Tisserand, Mecanique Celeste, vols. i. and iv. 
Hill, Collected Works, vol. ii 
Newcomb, American Joimial of Mathematics, vol. iii., and Astronomical 
Papers of the American Ephemeris, vol. v. 
Cowell, Monthly Notices of the Boyal Astrojiojiiical Society, vol. Ixix. 
The object of the present paper is purely practical ; it is to put in the 
hands of the computer a development which extends to any order for 
primary or secondary terms, and is substantially ready for use. A 
