(the old system of comet elements is here employed, as more convenient for our purpose) :—

The following are the collected values of , 8, i, reduced to the Equinox of 1910:—

The secular variation of i has been neglected altogether in the period now under discussion, as it appears to be too small to deduce with certainty, considering the roughness of the elements for all the apparitions before 1682.

(3) The co-ordinates of the planets referred to the cometary axes are assumed to repeat themselves at uniform intervals, but the interval for x', y' is taken slightly different from that for z', to allow for the adopted motions of the perihelion and the node.

In consequence of the change in the period and major axis of the comet's orbit, it is more convenient to use the ordinary

astronomical unit of length and the mean solar day as the unit of time. In our previous papers on the perturbations by Jupiter and Saturn 1835-1910 (M.N., lxvii. pp. 386, 511) every quantity tabulated was of zero dimensions in space and time.

(4) In the case of Jupiter and Saturn, the method of mechanical quadratures is used for values of u (the comet's eccentric anomaly) from o° to 90°, and from 270° to 360°. For the outer half of the orbit the definite integral method described in the above paper (p. 391) is employed; since the eccentricity of the comet's orbit is treated as constant, all the values on p. 409 can be used unchanged except those relating to the position and motion of the perturbing x y1 dx' I dy' planet; it may be noted that

a a na dť na dt

vary as

I

a

The portion of the perturbations in the outer half of the orbit that we have denoted by V4 (p. 391) is neglected altogether; this may make an error of a few days in the calculated period, but not sufficient to cause any doubt as to the identity of an observed comet.

The intervals of u for which the calculations are made are 41° for Jupiter and Saturn, 9° for Uranus, 18° for Neptune (the calculations for these two planets being made throughout the whole orbit), 1° for the Earth and Venus (the calculations for these two planets being made from the values of u 0° to 30°, 330° to 360°).

(5) We have omitted the discussion of the perturbation of e, 8, i, having merely found the quantities that come directly into the periodic time.

The following are the assumed co-ordinates of Venus and Earth, referred to the axes of the comet's orbit; they are deduced by a simple transformation from those given by Stratford in the Supplement to the Nautical Almanac for 1839, and are in units of the 3rd decimal: