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dn, the alteration in the mean motion, is equal to X sin uY cos u√1-e2 multiplied by a reducing factor, which is not introduced till after the summation; each value of dn is multiplied by the number of days to elapse before the next perihelion passage.

The following quantities are then formed for each value of u :— xY − yX, rX, r(xX+yY). In taking the sums of the columns, only half of the term at the top and bottom of each column is taken; the reason is obvious from the geometry of mechanical quadratures.

(dn

The factor for reducing dn to seconds of arc is

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The same factor multiplies the sum of dn x No. of days to next perihelion passage, and gives a quantity which we call A. The factor to multiply

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The factor to multiply(2X+yY) is 2 m'du: this gives a quantity which we call C.

Then

faç= A-faw √x - e2 - C.

And period in days is given by the expression

1296000 - far

n at beginning of revolution

du is the selected interval of u expressed in seconds of arc; thus for interval

1° du= 3600, for 41° du = 16200, for 9° du= 32400, for 18° du = 64800

log = 3'5563

log = 4'2095

log = 4'5105

log = 4.8116

The following are the assumed values of log m' for the different planets:

43893, +4°4781, 2 6*9799, h6·4558, H 5·6408, 57122; also

I

√1 − e2 = 2540, i.c. e '9672, log = 0.0145,

e

log

6п 365·256

= 8.7127.

The following tables exhibit the calculated quantities:

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"

Hence n at 1456 = 46°029

153146029+"0375-46" 404.

The last result is in good accord with de Pontécoulant, who gives (C.R., lviii. p. 708) 46" 410 for the value of n in 1531; as he has computed the perturbations by Jupiter and Saturn only for the revolutions 1531 to 1607, 1607 to 1682, his value is obviously not to be relied on beyond the first decimal.

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Hence for this revolution we have the equation—

1296000′′ - 3452"*3

n at 1378

28336 44, the observed period in days.

Hence n at 1378=45.614

1456=45.614+"0'407 = 46′′ 021.

There are now two independent determinations of n at 1456, which are satisfactorily near each other, considering the many abbreviations that we have introduced; we are justified in adopting their mean, viz. 46" 025. The corresponding values for 1378, 1531 are 45" 618, 46" 400 respectively; the close accord of the latter value with that of de Pontécoulant is satisfactory.

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We now proceed as follows: n at 1301 = 45" 618 - 0"'763

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This only differs 3d.5 from the value obtained by using Hind's identification of the comet of 1301, viz. 281414.07. The dis

cordance is so small that we are justified in accepting Hind's result with absolute confidence. The long discussion as to the identity of the comet of 1301 is thus happily ended. Dr. Galle, in his Cometenbahnen, p. 155, says of the comet of 1378, "Die erste als sicher zu betrachtende Erscheinung des Halley'schen Cometen." We may now claim to have brought the "sicher " returns one revolution further back, and to have thus obtained a firm startingpoint for further investigations, which have, in fact, already been commenced.

It will be remembered that Pingré, using the European observations alone, obtained elements quite unlike those of Halley; subsequently Laugier, using a combination of European and Chinese observations, obtained elements resembling those of Halley, except the position of the node, which was 90° greater. Finally, Hind rejected the European observations entirely, and showed that the Chinese ones could be well represented by the Halley elements. Our result indicates that he was justified in this course; it is rather a curious reversal of the present relations of European and Chinese astronomy. Many of Dr. Hind's older identifications rest on Chinese observations, and it is satisfactory to find that the accuracy of the latter in 1378 and 1301 is fully vindicated.

Having once satisfied ourselves of the identity of the comet of 1301, we may rewrite the last equation in the form

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"

2814107, the observed period in days.

Hence n at 1301 = 44'861

1378=44.861 +0" 763 = 45" 624.

Taking the mean of this and the previously adopted value for 1378, viz. 45" 618, we obtain 44" 858, 45" 620 for the adjusted values in 1301, 1378. It is advisable to make these successive adjustments in order to diminish cumulative error through quantities that we have neglected.

Observations of Comet d 1907, from photographs taken with the 30-in. Reflector of the Thompson Equatorial and the Astrographic 13-in. Refractor at the Royal Observatory, Greenwich.

(Communicated by the Astronomer Royal.)

The following positions of Comet d 1907 were obtained from photographs taken with the 30-in. reflector and the astrographic 13-in. refractor, with exposures of from 20 seconds to 3 minutes.

The plates were measured in the astrographic micrometer. Six reference stars were, as a rule, measured with the comet, their positions being derived from the Catalogues of the Astronomische Gesellschaft.

The positions given are not corrected for Parallax.
Log Parallax Correction log Parallax Factor

=

log A.

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