74 
Astronomical Society \ the Astronomer Royal 
differential coefficients, which are given by the observations, he 
arrives at t^^o equations in which the unknown quantities are p and 
^ (p denoting the comet’s distance from the earth). On elimina- 
d t 
ting an equation is found of the following form — 
C.p 
D 
(p°— Ep + F) 
1+ G, 
where C, D, E, F, G, are known numerical quantities. The solu- 
tion of this equation may be obtained with great facility (in respect 
of the general difficulties of the problem) by the method of trial 
and error ; and the author recommends, that in all cases which ad- 
mit of it, the equation be formed, and the solution found ; not only 
because the method is comparatively easy, but also because it is 
perfectly general, no assumption of parabolic, circular, or any other 
form of orbit, having been made. 
The author next proceeds to consider the cases in which the 
equation fails. These are, first, when the comet is in conjunction 
with, or in opposition to, the sun ; or when the sun, the earth, and 
the comet, are in the same straight line. Tn this case the first side 
of the equation becomes 0 divided by 0 ; and, as the two equations 
which involve the first differential coefficient of the comet’s distance, 
taken with respect to the time, also vanish in the same circum- 
stances, the failure is absolutely beyond remedy, and we can only 
wait until the comet is in a difierent part of its orbit. Secondly, 
the equation fails when the apparent path of the comet is directed 
to or from the sun’s place ; but in this case, the two equations in- 
volving the first differential coefficient of the distance do not neces- 
sarily fail; and, in fact, they cannot both fail, excepting under the sup- 
position of the first case ; therefore, by using one of them, or a new 
combination of them together with some new single assumption (as 
for instance, that the comet is moving in a parabola of unknown pe- 
rilielion distance), w^e may still determine the comet’s distance. 
Thirdly, the application of the equation may fail from causes con- 
nected with instrumental observations ; for as the second differential 
coefficients of the right ascension and declination both occur on the 
first side of the equation, and as these coefficients are affected by 
the whole of the errors of observations, which, if the interval be- 
tween the observations is short, receive very small divisors, any 
failure in the instrumental determination will produce a large error 
in their proportionate values. As it will sometimes occur that the 
observations made in declination are far more accurate than those 
made in right ascension, or vice versd^ in most cases one of the two 
equations which contain the comet’s distance and its first differential 
coefficient, will be preferable to the other; and the combination of 
this with the equation deduced from the assumption of a parabolic 
orbit, will lead to the elimination of the differential coefficient, and, 
consequently, give the distance. 
