The co-ordinates of the comet itself are found from the table (M.N., lxvii. p. 387) by multiplying, by a, using the special a value for each revolution, and the number of days from perihelion to each value of u by multiplying nt by period in days The co-ordinates of Jupiter are found from the table (M.Ν., lxvii. p. 394) by multiplying, a'a by 177. The following table gives the Julian days corresponding to certain values of g': The values for intermediate degrees may be found with sufficient accuracy by reckoning 12d per degree from the nearest table value. The co-ordinates of Saturn are found from the similar table, referred to (but not printed in detail), M.N., lxvii. p. 512. As the formulæ on which the table is based are given there, the table is not printed here. As the perturbations by Uranus and Neptune are always small, approximate co-ordinates suffice; those of Uranus are taken from de Pontécoulant (Mémoire sur le calcul des perturbations de la comète de Halley, 1829, pp. 37, 38, 48). They are as follows, in units of the first decimal: The perturbations produced by Neptune are still smaller than those of Uranus, and, the eccentricity of its orbit being small, it has been considered to move in a circle. The following are its co-ordinates at intervals of 6° of longitude, or 1003 days, in units of the first decimal : The co-ordinates recur with reversed signs after 30090 days. The dates of perihelion passage in 1378, 1456 were already known with accuracy from the observations; for that of 1301 Hind's value was taken as a working hypothesis, to be tested by the result; the Julian days of the various passages are given here for convenience of reference : The last value is uncertain by several days; the value given was that employed in our Jupiter and Saturn work. The following are the assumed co-ordinates of Jupiter and Saturn at the times given : The method pursued has been identical with that used by de Pontécoulant in the memoir quoted above. x - x, y y were found for each value of u; then (-x)2 + (y - y)2 + z^2 = p2, where p is the distance between the planet and comet. Then X, Y, the perturbing forces parallel to the axes, are proportional to |