182 ADDRESS DELIVERED BY THE PRESIDENT ON PRESENTING THE [579 
to remain of the highest degree of exactness for only a few centuries) to take not 
the mean values of the elements, but their values at a particular epoch during the 
period for which the theory is intended to be used. The adopted provisional elements 
of Neptune, and the elements of the disturbing planets, are accordingly not mean 
values, but values affected by secular and long inequalities, representing the actual 
values at the present time. Secondly, the form being decided on and the formulae 
obtained, the numerical values of the adopted provisional elements of the planet, and 
of the elements of the disturbing planets and their masses, have to be substituted, so 
as to obtain the actual formulae serving for the calculation of a provisional ephemeris; 
and such ephemeris, first of heliocentric, and then of geocentric positions, has to be 
computed for the period over which the observations extend. Thirdly, the ephemeris, 
computed as above, has to be compared with the observed positions ; viz. in the present 
case these are, Lalande’s two observations of 1795, and the modern observations at the 
Observatories of Greenwich, Cambridge, Paris, Washington, Hamburg, and Albai^, 
extending over different periods from 1846 to 1864: these are discussed in reference 
to their systematic differences, and they are then corrected accordingly, so as to reduce 
the several series of observations to a concordant system. In this way is formed a 
series of 71 observed longitudes and latitudes (1795, and 1846 to 1864); the comparison 
of these with the computed values shows the errors of the provisional ephemeris. 
Fourthly, the errors of the provisional elements have to be corrected by means of the 
last-mentioned series of errors: as regards the longitudes, the comparison gives a series 
of equations between 8e, Sn, 8h, 8k, and ¡i (correction to the assumed mass of Uranus). 
The discussion of the equations shows that no reliable value of yu, can be obtained 
from them; it indeed appears that, if Uranus had been unknown, its existence could 
scarcely have been detected from all the observations hitherto made of Neptune (far 
less is there any indication to be as yet obtained as to the existence of a trans- 
Neptunian planet): hence, finally, ¡jl is taken =0, and the equations used for the 
determination of the remaining corrections. As regards the latitudes, the comparison 
gives a series of equations serving for the determination of the values of 8p and 8q. 
And applying the corrections to the provisional elements, the author obtains his con 
cluded elements; viz. as already mentioned, these are the values, as affected by the 
long inequality, belonging to the epoch 1850. Fifthly, the tables are computed from 
the concluded elements, and the perturbations of the provisional theory. 
After the elements of Neptune were ascertained, the question of its action on 
Uranus was considered by Peirce in a paper in the Proc. American Acad., vol. i. 
(1848), pp. 334—337. This contains the results of a complete computation of the 
general perturbations of Uranus by Neptune in longitude and radius vector, but without 
any details of the investigation, or statement of the methods employed: it is accompanied 
by a comparison of the calculated and observed longitudes of Uranus (with three 
different masses of Neptune) for years at intervals from 1690 to 1845, and for one of 
these masses the residuals are so small that it appears that, using these perturbations 
by Neptune and Le Yerrier’s perturbations by Jupiter and Saturn, there existed a theory 
of Uranus from which quite accurate tables might have been constructed. But this 
was never done. The ephemeris of Uranus in the American Ephemeris was intended