In the computation of the parallaxes the adopted value of the Sun's mean horizontal parallax is 8" 80, and the geocentric distances A are taken from the Astronomische Nachrichten, Nos. 4191, 4194, and 4196.

Radcliffe Observatory, Oxford:

1908 May 30.

The Orbit of Jupiter's Eighth Satellite. By P. H. Cowell
and A. D. Crommelin.

In this paper the time is measured in units of 16 days from 1908 January 120 Paris mean time. The astronomical unit of length is used.

Paris mean time has been used because the Connaissance des Temps gives both heliocentric and geocentric longitudes and latitudes of Jupiter referred to the same equinox, the true equinox of date.

With 16 days as the unit of time,

It will be seen in the later parts of this paper that the Sun's disturbing force varies during the early part of 1908 between 10 and 6 per cent. of the attractive force of Jupiter, and is therefore by no means negligible. Moreover, if the attempt were made to express the co-ordinates of the satellite with the help of Delaunay's algebraical lunar theory, it would be found that Delaunay's quantities e, e and m take such large arithmetical values that Delaunay's series are insufficient for the purpose. Hence it is

best to calculate the orbit by quadratures, discarding the idea of an ellipse.

Let the jovicentric co-ordinates at time n+t referred to three rectangular directions be

Again, given the co-ordinates at any time, the accelerations can be calculated in accordance with Newton's law of gravitation. If X denote the x acceleration at time n+t,

The left-hand side of this last equation may be described in words as one-twelfth of the second difference of X, and it will be denoted for brevity by P.

Hence we have Xn+1 = Xn+ (xn− xn−1)+Xn+P2

with one error of 3a, which we shall assume to be negligible.

n-29

Now let us assume that -2, -1, n have been calculated, and Xn-2, Xn-1, Xn, and therefore also P P-1 We cannot at present calculate P,, for this requires a knowledge of X+1, and therefore of +1, Yn+1, n+1; but we may guess at the value of P with the help of the values of P-2, Pn-1 (or for the first few values of n, where P-2 has not been calculated, by trial and error); we then obtain a provisional value of +1 (and similarly yn+1, Zn+1); hence we calculate Xn+1, and so obtain a revised value of P1; the correction AP, is then added to +1, and in practice it is found that Xn+1, Yn+1, Zn+1 are unaffected."

We give a numerical application of the above method, taken from our calculations of a provisional orbit. In the illustration it is assumed that the orbit has already been calculated as far as t=5 (April 1); it is required to extend it to t=6 (April 17). For the sake of clearness, quantities already calculated are written without brackets, quantities guessed at are written in round brackets, and fresh quantities calculated accurately are enclosed in square brackets.

Hence by addition,

(x) = +0345 558, (y。) = −∙1674 443, (%) = Hence we calculate X, Y, Z, and hence

== -'0075 732.

[P] = +0000029, [Q] = + '000 014, [R5]=

'0000 020.

Hence applying the small differences between the true and provisional values of P5, Q5, R5,

[%]= +0345 560, [y]= − '1674 444, [%]= −0075 732.

The foregoing sufficiently explains how to calculate the motion from assumed initial positions; we now deal with the problem how to find an approximate orbit from the observations.

Retaining the previous notation, and interpolating a little among the observations, we shall assume as known the geocentric angular positions at three equidistant times n − 7, n, and n +T. Reasoning exactly as before we have

Xn+7 - 2Xn+ Xn-T

=

-X,72 with an error of 2074.

If we take the x direction at right angles to the geocentric directions of the satellite at times n T and n +7, then our ignorance of the geocentric distances at these times produces no uncertainty in the numerical values of + and -; we therefore have one unknown quantity only occurring, viz. the geocentric distance at time n; and the above equation, by trial and error, will determine this quantity.

The corresponding equations for y and z then determine the geocentric distances on the first and third occasions.

We may then revise our solution so as to correct for the residuals 204 etc.

In practice it is not necessary to take the x axis at right angles to the first and third geocentric distances; we shall then have to deal with three simultaneous equations for three unknown geocentric distances. The idea of resolution perpendicular to the outer geocentric distances is perhaps useful for forming a distinct mental conception of the process.

We shall not go into numerical details in connection with the method just sketched. It may be anticipated, however, that the solution is a dual one, or that two orbits which may be distinguished as direct and retrograde can be found to satisfy three observations. It unexpectedly turned out that the jovicentric distance on February 28 was less on the supposition of a direct orbit than for a retrograde orbit; but a very large eccentricity resulted from the former hypothesis. For this reason combined with the reasons stated in M.N., lxviii. p. 457, the retrograde solution was chosen as the one to be worked up. It will be seen that the corresponding orbit closely follows the observations. Now the dual solution implies that there are two distinct ways of making the coefficients of the squares of the time agree with observation. Of these two solutions the retrograde solution makes the coefficients of the third and fourth

2

powers of the time also agree with observation. Of course it
is not inconceivable that the direct solution might also give cube
and fourth terms agreeing with observation, but it is a priori
distinctly improbable. We have not, therefore, examined the direct
solution (or the question of a possible planetary solution) with any
minuteness. However closely a direct solution was found to fit
the observations, it would be impossible at the present moment to
reject the possibility that the retrograde solution was the right one.

The following table contains full details of our provisional orbit.
The axes are drawn as follows:- towards the first point of Aries
for 19080, y towards longitude 90°, z towards the pole of the ecliptic
xyz are the jovicentric co-ordinates of the satellite,

X, Y, Z1 are the accelerations of Jupiter on the satellite

1 1 1

X, Y, Z2

2 2

X Y Z are the sums of the three preceding quantities.
PQR are the twelfth parts of the second differences of X Y Z.

The values of P, Q, R for April 17 are enclosed in brackets to
indicate that their values have been inferred, and not calculated :
the values of x, y, z for May 3 are enclosed in brackets to indicate
that they depend upon the preceding values of P, Q, R.

Table exhibiting Details of Calculation of provisional Orbit of J. VIII.

-'0016771

-'0017204

+ '0021090

+ '0020670

+ '0021861 + '0024163
+ '0002237 + '0001093

+0019889

-'0020262

- '0000279
- '0018435
-0019368

+ '0020229

- '0019611
- '0000341
*0018823
+0018920

-'0007366-'0006392

+ '0016477 +'0005087 + '0004609 + '0004006 + '0000013 + '0000016

+ '0018634

-'0005159 + '0020967

+ '0000353
- *0003581

+ '0023472

+ '0000014 + '0000014 + '0000 15

'0000013 - '0000016

From the co-ordinates x, y, z of the preceding table, the tabular differences of geocentric right ascension and declination between the satellite and Jupiter were obtained. We exhibit also the interpolation for every fourth day.

Table for Paris Mean Noon, giving excess of Satellite's tabular R.A. and Decl. over that of Jupiter.

The next table gives the dates of the observations reduced to Paris time with the light times subtracted; the place of observation, and the observed geocentric distances in R.A. and declination of the satellite from Jupiter; also the excess of the corresponding calculated co-ordinates over the observed.