A CalciiIatio7i of the Orbit of the Comet. 107 



rattan in cometogTajihy, and I may say in astronomy, is 

 that of the periodical times of the comets. At first view, 

 it may appear surprising-, since the other elements are 

 attainable to a great degree of accuracy, that this should 

 yet be unknown. The same process, by which other 

 elements of 3. planefs orbit are determined, will also de- 

 termine that of their periodical times ; and why, it m.ay 

 be asked, does this not result from a similar process for 

 in\'estigating the orbit of a comet ? In answer to this, I 

 would observe, that the orbits of planets %^ary little from 

 circles, and consequently their periodical times may be 

 found nearly, by a comparison of their velocities with 

 that of any body moving in a circle about the center of 

 their orbits. The variation of their velocities, arising 

 from the deviation of their orbits from circles, may also 

 be determined ; as that deviation, in its incipient state, 

 or while the planet's orbit is an ellipsis, differing little 

 from a circle, is very great, compared with its effect in 

 respect to the perioclicai time, and is therefore suscep- 

 tible of determination, either from observations of the 

 planet's distance from the sun, or of its velocity. But 

 the orbit of a comet, is a very eccentric ellipsis, whereof 

 the deviation of curv^ature from that of a circle has arriv- 

 ed nearly to its limit ; and the variation of curvature 

 among ellipses of this sort, on which tlie proportion of 

 their axes, or of their periodical times depends, is so mi- 

 nute, as scarcely to be perceptible near the extremity of 

 their longer axes, or in the comet's orbits near the sun 

 and earth, where only they become visible to an observ- 

 er on the earth. For this reason, a parabolic orbit has 

 been assumed by astronomers, as sufficiently Ticcurate 

 for the calculation of every phenomenon, incident to a 

 comet's motion within the sphere of the planets. The 

 periodical time in a parabola, or an ellipsis, the ratio of 

 whose axes is hifimte, if I may use the expression, is 

 hifinite ; yet the curvature near the extremity of the 

 axes of such a figure, differs little from that of an ellip- 

 sis, whose axes are in a ratio of no great finite magnitude. 

 Thus, in an ellipsis, whereof the ratio of the axes is as 

 10 to 1, the difference of its parameter from that' of a 

 i parabola, the distance from the vertex being; the same, 



