74) Astronomical Society; the Astronomer Royal 



differential coefficients, which are given by the observations, he 

 arrives at two equations in which the unknown quantities are p and 



— ^ (p denoting the comet's distance from the earth). On elimina- 

 ting — an equation is found of the following form — 



C . p = -+ G, 



(p°— Ep + F)- 



where C, D, E, F, G, are knoAvn 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, M'hen 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. In this case the first side 

 of the equation becomes divided by ; 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 difi'ei'ent 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- 

 rihelion distance), we 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 versa, in most cases one of the two 

 equations which contain the comet's distance and its first differential 

 coefficient, Avill 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. 



