Royal Astronomical Society. 227 



tions of the problem, and it is, perhaps, impossible to recognise it 

 in the analytical expression without a much greater effort of the at- 

 tention than can be given when merely computing an orbit. With 

 good observations the method has the advantage of revealing any 

 obvious tendency to an ellipse or hyperbola ; and, besides, it will 

 in most cases afford a useful approximation to begin with in com- 

 puting the parabolic formulae. As to the expediency of putting the 

 observations to this preliminary test in all cases, there would, per- 

 haps, be little difference of opinion, if the labour of computation in 

 doing so were available in the last part of the process, and if the con- 

 ditions upon which the degree of accuracy depends could be easily 

 distinguished. 



" The object of this paper is to submit to the Astronomical So- 

 ciety an account of a method which has occurred to me of solving 

 the equation by means of a constant curve, and to show how the 

 preliminary calculation may be made available in Olbers's parabolic 

 method, and likewise in a differential method, without requiring the 

 original equatoreal position in either case to be transferred to the 

 ecliptic. The conditions of accuracy also become so apparent in 

 using this curve, that the effect of an error of right ascension or de- 

 clination may be estimated by inspection. 



" The method of solution is derived from the projection of the three 

 observations on the plane perpendicular to the direction of the motion 

 of the comet at the middle epoch. The earth's orbit being projected, 

 its deflection, caused by the sun's attraction, is brought into view, 

 and since its apparent direction is the same as that of the sun, and 

 the projected direction of the sun from the comet is the same as at 

 the earth, the radii vectores of both being identical on the projection ; 

 it is clear if the differentials at the middle time are alone considered 

 that the deflection of the orbit of the comet, as it appears on the 

 plane of projection, coincides in direction with the projected deflec- 

 tion of the earth's orbit, and that its magnitude depends on a function 

 of the angle at the comet. We thus obtain the means of forming an 

 equation for the angle at the comet in terms of the deflection 

 of the earth's orbit ; and this equation, although derived from a 

 simple geometrical construction, appears to be similar to that which 

 is given in the analytical discussion of the problem by Laplace, La- 

 grange, Legendre, and Airy. It depends wholly on the effect of the 

 sun's centripetal force during the elapsed time as it appears on the 

 plane of projection ; and, as this, in the short differential period of 

 a few days, bears but a small proportion to the projection of its chord, 

 or velocity, the results are much more liable to be affected by the 

 unavoidable errors of observation than if the equation expressed the 

 same unknown quantity in terms of the velocity. But in the last 

 case we have to suppose the nature of the conic section known, in 

 the first no assumption of the kind is required, the deflecting effect 

 of the sun's force being necessarily the same in all orbits at the same 

 central distance. 



" The equation for the angle at the comet is solved by drawing one 

 line on the constant curve, and the preliminary computation required 



R2 



