The method of calculating a parabolic orbit from three observations which we shall follow is the comparatively expeditious one proposed by Gibers, and demonstrated in his Abhandlung über die leichteste und bequemste Methode die Bahn eines Cometen zu berechnen, first published at Weimar in 1797, and since twice reprinted with considerable modifications and additions. The method is founded upon the principle that, if a c be the chord between the extreme positions of the comet in its orbit, and A C the similar chord of the earth’s path, the radii-vectores at the middle position cut a c and A C proportionally to the times occupied in describing the arcs, a supposition which, though not mathematically exact, is but little in error if the intervals between the observations are pretty nearly equal, and the arcs described small.

It may be convenient if the notation employed in the subsequent formulae be given here, at least as regards the principal quantities entering into our calculations. [table]

[table]

We suppose that the observations furnish three complete positions of the comet referred as usual to the equator, or expressed in right ascension and declination, with the mean times of observation at the respective places.

The first step will be to convert the observed right ascensions and declinations into longitudes (a) and latitudes (β), thus :— [table]

Thus we find α', a", a ", and β', β", β"', where the quantities with one accent apply to the first observation, those with two accents to the second place, and with three accents, to the last observation. This is to be understood throughout our formulae for the calculation of the orbit.

Now reduce the times of observation to the meridian of Greenwich by applying the longitude of the place of observation with its proper sign, and convert the times so reduced into decimals of a day; thus we have t ’ , t", t"'.

For each of these times interpolate from monthly page iii., in the Nautical Almanac, the sun’s longitude (A) and the logarithm of the earth’s radius-vector (R); the sun’s longitude in the Almanac being apparent, the amount of aberration (20" - 42 ÷ R), which is given in another part of the ephemeris, must be added to the apparent longitude, to obtain the true longitude required in the calculation. We have then A', A", N" and the logarithms of R', R", R'", and are ready to proceed with the application of Olbers’s method.

We commence by calculating Μ or , the ratio of the P

comet’s curtate distances from the earth at the first and third observations from [table]

The following equations must then be formed (fc is the chord of the comet-orbit between the extreme observations): [table]

If (f — i') be the .interval of time between the first and third observations we have, by Lambert’s theorem, [table]

With an assumed value for ρ' we calculate r, r ’ ", and ^k, and then t"' for comparison with the observed interval between the first and third observations, and vary p ’ in successive trials until the observed and calculated values agree. In this solution of the above equations by the method of trial and error, a first approximate value of ρ' may be inferred as follows :—

Writing the equation for V- [table]

The amount and direction of the error of interval between the extreme times of observation, resulting from this first value of p , , will, after a little experience, guide the computer to another value nearer to the true one; and the error-of the second assumption, compared with that of the first, again leads to a much closer value for the third approximation, and so on till the assumed value of ρ ’ produces an agreement between the calculated and observed intervals. In practice we have not found any great advantage on adopting one or other of the devices suggested for obtaining successive values of ρ' by use of tables or otherwise—the simple method of continued approximation, by deducing a new value of the curtate distance proportional to the errors in the two preceding assumptions, will be found in the great majority of cases sufficiently expeditious and as little troublesome as any other.

In working Lambert’s equation, proceed as follows :— [table]

The approximations to ρ' may be continued until z' - z" agrees with (½" - i'), within 2 or 3 in the fifth place of decimals, though if the computer has only rough observations at command, à larger error may be tolerated.

The comet’s curtate distance from the earth at the third observation is given by

ρ'"=M∕

With the final values of √, r"', ρ' and p"', the direct calculation of the elements of the orbit commences.

The heliocentric longitudes F, θ''', and latitudes λ', λ'", are obtained from [table]

in which equations the right-hand quantities are known.

The values of √ and r" ’, resulting from these equations, should agree with the preceding ones if the calculations have been correctly performed. This agreement forms the first verification of the work.

If θ"' is in advance of ff, the motion in the orbit is direct; if the contrary be the case, the motion is retrograde.

Then, if the motion be direct, the longitude of the ascending node ’ ( ß ) and inclination of the orbit to the ecliptic (i) will be found from [table]

and if the motion be retrograde from [table]

The distances of the comet from the ascending node reckoned upon the orbit, at the first and third observations (√, w"), are given in the case of direct motion by [table]

or, if the motion be retrograde, by [table]

The arc u" ’- u is equal to the difference of true anomalies, and the true anomaly at the first observation (√) will be obtained from [table]

or from [table]

The perihelion distance (<?) = /· cos. 2 ¾υ = r'. cos. 2 ?v .

The longitude of the perihelion, reckoned on the ecliptic to the node and thence on the orbit, is given by [table]

As a further verification of the calculations, we have— [table]

which should give the former value of l·.

We have now only to determine the time of perihelion passage (T), by finding the interval between the first observation and the perihelion (τ) from υ' and q, by means of the equation— [table]

Similarly we may find the interval from the third observation to perihelion by substituting υ"' for v' ; the times thus separately determined should agree, and this agreement will afford a third check upon the accuracy of our work.

Thus the whole of the elements of the parabolic orbit are found, and it is always desirable to ascertain how the geocentric place calculated from these elements for the time of the second observation agrees with the position observed; the first and third places are necessarily represented.

In the computation of a geocentric position from parabolic elements we may proceed thus :—

Find the interval from perihelion passage to the time for which we require to compute (t - T), in days and decimals. [table]

We have thus the true anomaly and radius-vector. Then, if the motion be direct,— [table]

or, if the motion be retrograde,— [table]

—equations which give the heliocentric longitude and latitude (θ, λ). The geocentric longitude and latitude (a, β) and the true distance from the earth (Δ) are then obtained from— [table]

If the position of the comet as referred to the equator is required,—. [table]

Then

_ . eos. (N + e)

tan. R. A. — - c03 y — .tan. α tan.. Decl. = tan. (N + e). sin. R. A

As an example of the calculation of the orbit of a comet by Olbers’s method, we will compute the elements of the comet discovered at the Observatory of Marseilles by Μ. Borrelly on the 6th of December 1874, employing three observations taken at that Observatory on December 7, 16, and 26, viz. :— [table]

Converting the right ascensions into arc, and the times into decimals of a day, after subtracting 21 m 35" from them, for reduction to the meridian of Greenwich, we have— [table]

The obliquity of the ecliptic, from the Nautical Almanac, was 23° 27' 28", and hence, by the formulae p. 183, we find the following positions referred to the plane of the ecliptic, and interpolating for the above times from the same ephemeris the corresponding longitudes of the sun and log. radii-vectores of the earth, correcting the sun’s longitudes for aberration, and reducing all to mean equinox of 1875 0. [table]

The right ascension and declination are thus converted into longitude and latitude for the first observation :— [table]

And so for the second and third positions.

The interpolation of the sun’s longitudes and the log. radii-vectores of the earth from monthly page iii. of the Nautical Almanac requires no illustration.

We now form the angles α' - A", a" - A", a"- A", <tc., and take out the sines and cosines required; and it is always convenient to have these functions and other of the principal quantities copied in plain figures on a paper separate from the calculations. Thus we have,— [table]

We have— [table]

flf

The calculation of Μ = ^~τ , is as follows, by (II.) :— P [table]

Next, we form the equations (III.) for the determination of r' 2 , r"' 2 , and ∕c 2 by successive assumptions for the value of p',— [table]

Therefore the equations for r' 2 and r"' 2 are— [table]

So that the equation for F i is thus formed,— [table]

And thus substituting logarithms in the factors for p' and p' 2 , our equations stand thus, in the form for proceeding with the work— [table]

We have now to find the value of p' by trial and error. Here, if = F + G. o' + Hp' 2 , we have— [table]

And for a first approximate value of p', by (IV.),— [table]

In the earlier approximations we may use five-figure logarithms. With p' = 0'649, the work proceeds thus— [table]

If for a second approximation we take p' = 0'7139, and calculate z ’ — z" precisely as before, the error in the interval from the first to the third observation, or (i''' - i'), is found to be - 0 d '3606, which, compared with the error of the first assumption ( - 2 d '0488), shows a change of + l d, 6882 for an increase of 0O649 in p; or of 1 ⅛th part, and by mere proportion we have p = 0'72776, for a third approximation, giving the error in interval = + 0 d O222, so that we are now approaching the true value, and with p' = 0*7269562, obtained from the errors of the second and third trial in the same way that the third value of si was inferred, we may substitute seven-figure logarithms and work more closely; it will thus be found that the error in interval corresponding to the fourth assumption for p' is reduced to +0 d O0l67, or less than 2∣ minutes, and if we are only seeking an approximate knowledge of the orbit, the direct calculation of the elements might proceed with this fourth value of ρ'. However, to make the computation in this example a little more complete, we work out two further hypotheses, and finally adopt for the correct value of ρ'...0*7268994, with which the calculation is as follows :— [table] [table] [table]

which may be considered a perfect agreement.

We have thus for the direct calculation of the orbit Log. ρ'...9*8614743, and, since p'" = Mp', we find Log. p"'...9*5314543. Log. r'...0*0708694, and Log. ∕"... 0*1742799, as given by our final approximation.

With the aid of the formula (VI.) the heliocentric longitudes and latitudes of the comet at the first and third observations are found as follows :— [table]

so that our first verification is complete.

The comet’s heliocentric longitude at the third observation (0"') being less than that at the first observation (0'), the motion in the orbit is retrograde or contrary to the order of the signs, and we therefore proceed to determine the longitude of the ascending node (β) and the inclination sslbv the second set of e∩uations in (VITA ∙ thus— [table] [table]

Then for the arguments of latitude at first and third observations (u, u"') — [table]

.∙. √"-u'=10° 35'2"O, and i(u"'-√)-5 c 17'31" 0 .

We have now to calculate the true anomaly at the first observation from the radii-vectores r', r ’ and the included angle u -u, which is ≈v'"-v', and for this purpose will employ both expressions for tan. ⅜z> in (IX.)— [table]

The perihelion distance (<7) = r'. cos. 2 ⅜√, or [table]

The longitude of the perihelion (ττ)— [table]

It remains only to determine the time of perihelion from (XII.), computing both from υ and √" = √ + (√"-√) = 109° 34'45"O, so as to have the final verification of the work— [table]

We have now the whole of the elements of the parabolic orbit, viz., as usually written and entered in catalogues,— [table]

As already remarked, it is always desirable to ascertain how the comet’s geocentric position, calculated from the elements thus obtained, agrees with the observed position. A close agreement where good observations have been employed, of course, indicates that the real path of the comet in space does not much differ from a parabola, while a considerable difference, iae., one exceeding the probable error of the observation, may be due to the ellipticity of the orbit, and the comet may prove to be one of no long period. We will, therefore, proceed to compute the longitude and latitude from the above elements for the time of the second observation. [table]

Instead of using Barker’s table, we will compute the true anomaly directly by the formulae (XIII.); thus, [table]

We then obtain the comet’s heliocentric longitude on the ecliptic (0") and heliocentric latitude (λ"), the motion in the orbit being retrograde, from equations (XV.) :— [table]

[table]

So that the errors of elements for the second observation may be expressed in transcription thus :— [table]

These errors are not greater than may be looked for, in a computation upon the method we have adopted.

We have computed the true distance of the comet from the earth at the second observation Δ". If the true distances at the first and third observations are desired, we have Δ' = p , Δ'" = - p n ,,, , or, in the present case, cos. β cos. β

Δ' = 1’28437, Δ"'= 1 ’ 24574, so that the comet was slowly approaching the earth during the interval over which the observations extend.

If it be preferred to compare with the observed right ascension and declination, the formulae (XVII.) have yet to be applied, the calculation, as will be seen, being very similar to that in the conversion of right ascension and declination into longitude and latitude.

[The formation of the equations for the determination of p', p"', and k will perhaps be found the most slippery part of the computation by the beginner, and we add therefore two or three sets of data from observation and the ephemeris, with the resulting equations, which may be verified for the sake of obtaining a better acquaintance with this part of the work. [table]

The equations are [table] [table]

The equations are [table] [table]

The equations are [table]

It will be remarked that the apparent motion of this comet was very slow during the interval we have taken; it afforded a case where the orbit could only be improved by increased length of observation.]

We may correct the elements thus obtained for the main effect of the error due to the assumption made on commencing our calculation, by the following process, also suggested by Olbers, and applicable to the same observations.

There are already found r', r"', v, v" t and p'.

τ , i P". sin. (√"-√') (i"'-A , R'". sin. (A ff '-A")

u p r' . sin. (√' -v')~(i" -t') an ^=R' . sin. (A" -A') sh t ' ~ t } m = . tan ∕,^ , as in the calculation of the (i"-i')∙ m sin. (a -A")

ratio of the curtate distances fa or Μ.

P

Then compute N from [table]

Multiply those terms in the equations for p'" 2 and λ 2 which contain Μ by H , and the term in the equation for p'" 2 which contains Μ 2 by H 2 ; the equation for p' 2 not containing Μ or M 2 remains unchanged. With this new system of equations we find corrected values of ρ' and of r' and r'", and the elements of the orbit therefrom as before, p'" is now Μ . H . f .

To apply the above formulae to our preceding example we have [table]

[table]

With this corrected value of log. Μ, we might recalculate the co-eflìcients of p' and p' 2 in the equations for r"' 2 and k 2 , and complete the calculation of the orbit, but as the method of procedure is precisely that already illustrated by an example, it is unnecessary to occupy space here by so doing.

In the majority of cases in practice, the first elements of a comet’s orbit are calculated from a much shorter interval of observation than has been taken in the preceding example,—not infrequently from observations on consecutive nights, and in such cases our elements may be open to considerable correction, though the natural desire of the astronomer to learn something of a new comet’s position in the system, its track in the heavens, or possible identity with a comet already calculated, induces as speedy a determination of the orbit, however approximate, as practicable.

If the observations used in the first computations are near together, or the geocentric motion is slow, it will be preferable to wait for later positions, rather than occupy time in attempting a closer representation of the middle place. When later observations are available, the orbit ∕∕∕

may be re-calculated, Μ or v- , being determined by 01- bers’s formulae of correction, employing r', √", p ’, &c., as deduced from the first orbit. But the following general method of correcting approximate elements of a parabolic orbit, which has been widely used, will be found as satisfactory, though requiring great care in working. It is generally known as the method of variation of curtate distances.

We select three good observations at as wide intervals as practicable. These observations should be corrected for the effects of parallax aιid aberration by means of distances from the earth (Δ) calculated from approximate elements.

The aberration will be most conveniently taken into account by subtracting 497 β '8 × Δ from the time of observation, and interpolating the values of A and Log. R from the Nautical Almanac for the time thus reduced.

Then, introducing the observed longitude and latitude and the value of p, calculated from the approximate orbi⅛ find θ, λ, and r at the first and third observations from— [table]

From θ', λ', f, and θ'", λ"', r"', we compute the elements in the same manner as before, and thence the geocentric longitude for the time of the second observation, which call α 1 . (The geocentric latitude may be substituted for the longitude, if it be changing more rapidly.) Also find the time by these elements between the first and third observations, which call i 1 . Now vary p' by a small quantity, as 0 , 01 or 0'005 ( = m), and find 0', λ', r' again, and with these new values, combined with the previous ones for λ"', r"', coinpute the elements, and compare again with the second longitude, and call the difference from the longitude first computed r; also find the interval between the first and third observations,...