u153It appears, then, that the velocity of comets (p. 471), so far as it can be determined by these rude ways of computing, is that very velocity with which parabolas, or ellipses near to parabolas, ought to be described; and therefore the distance between a comet and the sun being given, the velocity of the comet is nearly given. And hence arises this problem.

PROBLEM.

The relation betwixt the velocity of a comet and its distance from the sun's centre being given, the comet's trajectory is required.

[Pg 567]

If this problem was resolved, we should thence have a method of determining the trajectories of comets to the greatest accuracy; for if that relation be twice assumed, and from thence the trajectory be twice computed, and the error of each trajectory be found from observations, the assumption may be corrected by the Rule of False, and a third trajectory may thence be found that will exactly agree with the observations. And by determining the trajectories of comets after this method, we may come, at last, to a more exact knowledge of the parts through which those bodies travel, of the velocities with which they are carried, what sort of trajectories they describe, and what are the true magnitudes and forms of their tails according to the various distances of their heads from the sun; whether, after certain intervals of time, the same comets do return again, and in what periods they complete their several revolutions. But the problem may be resolved by determining, first, the hourly motion of a comet to a given time from three or more observations, and then deriving the trajectory from this motion. And thus the invention of the trajectory, depending on one observation, and its hourly motion at the time of this observation, will either confirm or disprove itself; for the conclusion that is drawn from the motion only of an hour or two and a false hypothesis, will never agree with the motions of the comets from beginning to end. The method of the whole computation is this.

LEMMA I.

To cut two right lines OR, TP, given in position, by a third right line RP, so as TRP may be a right angle; and, if another right line SP is drawn to any given point S, the solid contained under this line SP, and the square of the right line OR terminated at a given point O, may be of a given magnitude.

It is done by linear description thus. Let the given magnitude of the solid be M2 × N; from any point r of the right line OR erect the perpendicular rp meeting TP in p. Then through the point Sp draw the line Sq equal to . In like manner draw three or more right lines S2q, S3q, &c.; and a regular line q2q3q, drawn through all the points q2q3q, &c., will cut the right line TP in the point P, from which the perpendicular PR is to be let fall. Q.E.F.

[Pg 568]

By trigonometry thus. Assuming the right line TP as found by the preceding method, the perpendiculars TR, SB, in the triangles TPR, TPS, will be thence given; and the side SP in the triangle SBP, as well as the error . Let this error, suppose D, be to a new error, suppose E, as the error 2p2q 3p3q to the error 2p3p; or as the error 2p2q D to the error 2pP; and this new error added to or subducted from the length TP, will give the correct length TP E. The inspection of the figure will shew whether we are to add to or subtract; and if at any time there should be use for a farther correction, the operation may be repeated.

By arithmetic thus. Let us suppose the thing done, and let TP + e be the correct length of the right line TP as found out by delineation; and thence the correct lengths of the lines OR, BP, and SP, will be , BP + e, and Whence, by the method of converging series, we have , &c., = , &c. For the given co-efficients , , putting , , , and carefully observing the signs, we find , and . Whence, neglecting the very small term , e comes out equal to -G. If the error is not despicable, take .

And it is to be observed that here a general method is hinted at for solving the more intricate sort of problems, as well by trigonometry as by arithmetic, without those perplexed computations and resolutions of affected equations which hitherto have been in use.

LEMMA II.

To cut three right lines given in position by a fourth right line that shall pass through a point assigned in any of the three, and so as its intercepted parts shall be in a given ratio one to the other.

Let AB, AC, BC, be the right lines given in position, and suppose D to be the given point in the line AC. Parallel to AB draw DG meeting BC in G; and, taking GF to BG in the given ratio, draw FDE; and FD will be to DE as FG to BG. Q.E.F.

[Pg 569]

By trigonometry thus. In the triangle CGD all the angles and the side CD are given, and from thence its remaining sides are found; and from the given ratios the lines GF and BE are also given.

LEMMA III.

To find and represent by a linear description the hourly motion of a comet to any given time.

From observations of the best credit, let three longitudes of the comet be given, and, supposing ATR, RTB, to be their differences, let the hourly motion be required to the time of the middle observation TR. By Lem. II, draw the right line ARB, so as its intercepted parts AR, RB, may be as the times between the observations; and if we suppose a body in the whole time to describe the whole line AB with an equal motion, and to be in the mean time viewed from the place T, the apparent motion of that body about the point R will be nearly the same with that of the comet at the time of the observation TR.

The same more accurately.

Let Ta, Tb, be two longitudes given at a greater distance on one side and on the other; and by Lem. II draw the right line aRb so as its intercepted parts aR, Rb may be as the times between the observations aTR, RTb. Suppose this to cut the lines TA, TB, in D and E; and because the error of the inclination TRa increases nearly in the duplicate ratio of the time between the observations, draw FRG, so as either the angle DRF may be to the angle ARF, or the line DF to the line AF, in the duplicate ratio of the whole time between the observations aTB to the whole time between the observations ATB, and use the line thus found FG in place of the line AB found above.

It will be convenient that the angles ATR, RTB, aTA, BTb, be no less than of ten or fifteen degrees, the times corresponding no greater than[Pg 570] of eight or twelve days, and the longitudes taken when the comet moves with the greatest velocity; for thus the errors of the observation will bear a less proportion to the differences of the longitudes.

LEMMA IV.

To find the longitudes of a comet to any given times.

It is done by taking in the line FG the distances Rr, , proportional to the times, and drawing the lines Tr, . The way of working by trigonometry is manifest.

LEMMA V.

To find the latitudes.

On TF, TR, TG, as radiuses, at right angles erect Ff, RP, Gg, tangents of the observed latitudes; and parallel to fg draw PH. The perpendiculars rp, , meeting PH, will be the tangents of the sought latitudes to Tr and as radiuses.

PROBLEM I.

From the assumed ratio of the velocity to determine the trajectory of a comet.

Let S represent the sun; t, T, , three places of the earth in its orbit at equal distances; p, P, , as many corresponding places of the comet in its trajectory, so as the distances interposed betwixt place and place may answer to the motion of one hour; pr, PR, , perpendiculars let fall on the plane of the ecliptic, and the vestige of the trajectory in this plane. Join Sp, SP, , SR, ST, tr, TR, , TP, and let tr, , meet in O, TR will nearly converge to the same point O, or the error will be inconsiderable. By the premised lemmas the angles rOR, , are given, as well as the ratios pr to tr, PR to TR, and to . The figure [Pg 571] is likewise given both in magnitude and position, together with the distance ST, and the angles STR, PTR, STP. Let us assume the velocity of the comet in the place P to be to the velocity of a planet revolved about the sun in a circle, at the same distance SP, as V to 1; and we shall have a line to be determined, of this condition, that the space , described by the comet in two hours, may be to the space (that is, to the space which the earth describes in the same time multiplied by the number V) in the subduplicate ratio of ST, the distance of the earth from the sun, to SP, the distance of the comet from the sun; and that the space pP, described by the comet in the first hour, may be to the space , described by the comet in the second hour, as the velocity in p to the velocity in P; that is, in the subduplicate ratio of the distance SP to the distance Sp, or in the ratio of 2Sp to SP + Sp; for in this whole work I neglect small fractions that can produce no sensible error.