Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 9)

579] GOLD MEDAL OF THE SOCIETY TO PROFESSOR SIMON NEWCOMB. 
181 
he observed, however, that this failure is accounted for by an accidental mistake in 
the computation of the perturbations of the radius vector by Jupiter. 
Professor Newcomb’s theory of Neptune is published in the Smithsonian Contributions 
under the title “ An Investigation of the Orbit of Neptune, with General Tables of its 
Motion,” (accepted for publication, May 1865). The errors of the published ephemerides 
were increasing rapidly; in 1863 Walker’s was in error by 33", and Kowalski’s by 
22"; both might be in error by 5' before the end of the century. The time was come 
when (the planet having moved through nearly 40°) the orbit could be determined 
with some degree of accuracy. The general objects of the work are stated to be: 
(1) To determine the elements of the orbit of Neptune with as much exactness 
as a series of observations extending through an arc of 40° would admit of. 
(2) To inquire whether the mass of Uranus can be concluded from the motion 
of Neptune. 
(3) To inquire whether these motions indicate the action of an extra-Neptunian 
planet, or throw any light on the question of the existence of such planet. 
(4) To construct general tables and formulae, by which the theoretical place of 
Neptune may be found at any time, and more particularly between the years 1600 and 
2000. 
The formation of the tables of a planet may, I think, be considered as the 
culminating achievement of Astronomy: the need and possibility of the improvement 
and approximate perfection of the tables advance simultaneously with the progress of 
practical astronomy, and the accumulation of accurate observations; and the difficulty 
and labour increase with the degree of perfection aimed at. The leading steps of the 
process are in each case the same, and it is well-known what these are; but it will 
be convenient to speak of them in order, with reference to the present tables: they 
are first to decide on the form of the formulae, whether the perturbations shall be 
applied to the elements or the coordinates—or partly to the elements and partly to 
the coordinates; and as to other collateral matters. These are questions to be decided 
in each case, in part by reference to the numerical values (in particular, the ratios 
and approach to commensurability of the mean motions), in part by the degree of 
accuracy aimed at, or which is attainable—the tables may be intended to hold good 
for a few centuries, or for a much longer period. The general theory as regards these 
several forms ought, I think, to be developed to such an extent, that it should be 
possible to select, according to the circumstances, between two or three ready-made 
theories; and that the substitution therein of the adopted numerical values should be 
a mere mechanical operation ; but in the planetary theory in its present state, this is 
very far from being the case, and there is always a large amount of delicate theoretical 
investigation to be gone through in the selection of the form and development of the 
algebraical formulae which serve as the basis of the tables. In Prof. Newcomb’s theory 
the perturbations are applied to the elements; in particular, it was determined that 
the long inequality arising from the near approach of the mean motion of Uranus to 
twice that of Neptune (period about 4,300 years), should be developed as a perturbation, 
not of the coordinates, but of the elements. And it was best, (as for a theory designed
	        
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