180 
ADDRESS DELIVERED BY THE PRESIDENT ON PRESENTING THE [579 
coordinates x, y, z, as periodic functions of the time. With these twelve elements, the 
expressions for the variations assume the canonical form 
dk e _ dR de 0 _ dR . 
dt de 0 ’ dt dk e ’ 
The concluding part of the Memoir contains approximate calculations which seems 
to show that the whole process is a very practicable one: but the author remarks that 
it is only doing justice to Delaunay to say that, starting from his (Delaunay’s) final 
differential equations, and regarding the planet as adding new terms to the disturbing 
function, there would be obtained equations of the same degree of rigour as those of 
his own Memoir. 
Everything in the Lunar Theory is laborious, and it is impossible to form an 
opinion as to the comparative facility of methods; but irrespectively of the possible 
applications of the method, the Memoir is, from the boldness of the conception and 
beauty of the results, a very remarkable one, and constitutes an important addition to 
Theoretical Dynamics *. 
I come now to the planets Neptune and Uranus: it is well-known how, historically, 
the two are connected. The increasing and systematic inaccuracies of Bouvard’s Tables 
of Uranus were found to be such as could be accounted for by the existence of an 
exterior disturbing planet; and it was thus that the planet Neptune was discovered by 
Adams and Le Verrier before it was seen in the telescope, in September 1846. It was 
afterwards ascertained that the planet had been seen twice by Lalande, in May 1795. 
The theory of Neptune was investigated by Peirce and Walker: viz. Walker, by means 
of the observations of 1795, and those of 1846—47, and using Peirce’s formulae for the 
perturbations produced by Jupiter, Saturn, and Uranus, determined successfully two sets 
of elliptic elements of the planet. The values first obtained showed that it was 
necessary to revise the perturbation-theory, which Peirce accordingly did, and with the 
new perturbations and revised normal places, the second set of elements (Walkers 
Elliptic Elements II.) was computed. With these elements and perturbations there was 
obtained for the planet from the time of its discovery a continuous ephemeris, published 
in the Smithsonian Contributions, Gould’s Astronomical Journal, and since 1852 in the 
American Ephemeris and the Nautical Almanac. The theory was next considered by 
Kowalski in a work published at Kasan in the year 1855. The long period inequalities 
are dealt with by him in a manner different from that adopted by Peirce, so that 
the two theories are not directly comparable, but Professor Newcomb, by a comparison 
of the ephemerides with observation, arrives at the conclusion that the theory of 
Kowalski (although derived from observations up to 1853, when the planet had moved 
through an arc of 16°) was on the whole no nearer the truth than that of Walker; 
* Since the above was written, Professor Newcomb has communicated to me some verj* interesting details 
as to the extent to which he has carried his computations, and in particular he mentions that, considering 
the action of each planet from Mercury to Saturn, he has (in regard to the terms the coefficients of which 
might become large by integration) estimated the probable limiting value of more than fifty such terms of 
period from a few years to several thousands without finding any which could become sensible, except the 
term leading to Hansen’s first inequality produced by Venus.