15 
on the determination of the Orbits of Comets. 
Among the various changes to which the comet’s apparent path 
is subject, and of which an arbitrary choice may be made, for the 
purpose of determining the distance in the cases in which the 
general equation fails, or becomes unsafe, the author considers the 
following to be the best : — viz. first, the curvature of the comet’s 
path, produced by the sun’s action (or the deflection measured only 
in the direction perpendicular to the apparent path) ; second, the 
acceleration in its path, produced by the sun’s action (or the deflec- 
tion measured only in the direction of its path) ; third, the deflection 
in the direction in which both the sun’s action on the comet and the 
sun’s action on the earth would cause a change of the comet’s appa- 
rent place (or the deflection measured along the great circle joining 
the comet with the sun). These changes are severally considered, 
and the method of forming the equation proper for each condition 
explained, and rules deduced for the guidance of the computer in all 
the particular cases in which the direct method cannot be followed. 
In these investigations the correction of observed places of the comet 
for parallax is entirely omitted, as it is most convenient, when p is ap- 
proximately found, to correct the observations for the corresponding 
parallax, to make the proper alteration in the second differential co- 
efficients, and then to repeat the process of approximation to the 
value of p. 
Having given the methods for finding the distance and its dif- 
ferential coefficient, the author concludes his first section with an 
indication of the process by which the elements of the orbit are 
computed. In the rules for the selection of the equations on the 
parabolic assumption, some considerations are introduced which are 
new and important. 
The second section contains remarks on the method of obtaining 
numerical values of the differential coefficients of the right ascen- 
sion and declination from the observations. In the use of these 
quantities, what we have to consider is, not the effect of absolute 
error in their values, but of proportional error. An error of a single 
second in the value of the second differential coefficient of Al may 
produce an ultimate error as great as would be produced by twenty 
seconds in the value of the first differential coefficient ; or as great 
as would be produced by ten minutes in the Al itself. This con- 
sideration allows the computer to determine many of the numbers 
which enter into the equations after the second observation : the 
method of proceeding is as follows : — 
“ Adopt for the epoch the middle time between the first and 
second observations : then the first differential coefficients of a and 
/3 (a denoting the right ascension, (3 the declination) will be obtained 
accurately by dividing the changes of a and (3 by the intervening 
time ; and the values of a and j3 for the epoch will be obtained 
with sufficient accuracy, by taking the means of a and (3 for the two 
observations.” 
The third and last section of the Memoir gives practical rules 
for the computation of the observations. The successive steps of 
the process, from the first observations to the determination of the 
