On a casual inspection of the results the amount of the value of R-D in the neighbourhood of the pole is remarkable, and appears to be abnormal, and it would appear as if no simple formula would express the law of the difference. To exhibit this more accurately, however, the observations have been grouped according to their closeness in N. P. D. and the means taken.

The following Table exhibits the results:

Excess of Reflexion-Result above Direct-Result from Groups of Stars.

No very intelligible or simple law is seen in these numbers, except that a constant additive correction to the D observations and a negative correction to the R observations seem to be required, and probably an additional additive correction to the D observations for north stars, and subtractive correction for south stars (the same corrections with signs changed being applied to the R observations) depending on the zenith distances.

If we assume that the values of R-D can be expressed by the formula,

+0" 96 1"-68 x Sin Zenith Distance.

then, the correction to reflexion observations is

- o":48+0"·84 × Sin Z. D.

and, the correction to direct observations is

+o"480"·84 × Sin Z. D.

and these corrections have been provisionally adopted.

The stars given in the following Table were found available for the correction of the assumed colatitude, and the results need very little explanation, except that the weights are proportioned to the numbers derived from the expression 2(p+1)nn'

n + n'

,p being the number

of stars below that for which each is calculated, and n and n' the number of observations above and below the pole.

If z be the correction to the colatitude, we shall have, by dividing the sum of the products by the sum of the weights,

2z+1°30=0,

or, the correction to the assumed colatitude is

o′′·65.

This result is substantiated by that of the year 1861 based on observations of 1861, and I have therefore adopted it for the year 1862; that is, all the results have been diminished by

true colatitude being assumed to be 38° 14′ 24′′ 15.

o"65, the

On account, however, of the difficulty attending the R-D investigation, the result is still liable to uncertainty.

It may be remarked that the corrections applied for discordance of direct and reflexion observations, and for colatitude, harmonize very satisfactorily all the observations made of stars in the neighbourhood of the pole.

SECTION VII.-Catalogue of Concluded Mean R.A.'s and Mean N.P.D.'s for 1862, Jan. 1, of Stars observed in the Year 1862; with the Annual Precessions.-Pages 67 to 93.

Some of the columns require no explanation, and notice will be taken only of those which require it.

The magnitudes of the stars are in general those given in the Separate Results for R.A., except in cases wherein the magnitude of a star has been noted on an evening in which the N. P. D. only has been observed.

The Mean Right Ascensions for 1862, Jan. 1, are the means of those given in the Separate Results for R.A., without any alteration, excluding those results inclosed within brackets.

The Precessions in R. A. for stars in the Nautical Almanac List are taken from that work, and therefore in general include proper motion. These precessions are marked with an asterisk. For stars included in the British Association Catalogue, the geometrical precessions there given are set down, due regard being had to secular variation in bringing up the right ascensions to 1862. The precessions for all other stars are computed with the constants given in the Tabula Regiomontana.

The same explanation will serve for the Precessions in N. P.D., excepting that none of them needed computing, since the precession being a function of R.A. only, they are easily taken from the B.A.C.

The Mean North Polar Distances for 1862, Jan 1, are the means of those given for each star in the Section of Separate Results for N. P.D., corrected for discordance of direct and reflexion results, and for error of colatitude, as investigated in the preceding Section of the Introduction.

RADCLIFFE OBSERVATIONS, 1862.

g

The notes at the end of the Catalogue are the result of a very careful scrutiny and comparison with the B. A. C. and other Catalogues, and serve to show that Lacaille's right ascensions are in general greatly in error.

SECTION VIII.-Horizontal and Vertical Diameters, Right Ascensions, and North Polar Distances of the Centres of the Sun, Moon, and Planets; and R.A. and N.P.D. of Comet II of 1862; (the N.P.D.'s corrected for Discordance of Direct and Reflexion, for Error of Colatitude, and for Flexure of the Telescope of the Transit Circle): compared with the Nautical Almanac.-Pages 94 to 105.

Diameters.-The "Observed Duration of Transit" of the Sun's Diameter is obtained by taking the differences of the times of transit of the first and the second limb, and the "Seconds of Nautical Almanac" are obtained from the Nautical Almanac by doubling the "Sidereal Time of the Semi-diameter passing the Meridian" given in that work. The large positive tabular errors observable in the results are mostly those observed by Mr. Lucas (whose observations may be inferred from the signatures appended to the table of clockerrors and to the Separate Results for Mean N. P. D.), and by them. it would appear that his estimation of the time of transit of the first limb over the wires is different from that of the second limb. For cases in which only one limb has been observed the Nautical Almanac value of the "Time of Transit of Semi-diameter" has been adopted without alteration.

The same explanation will suffice for the Vertical Diameters of the Sun, excepting that these are corrected for refraction. The mean of thirty-six values of the errors of Vertical Semi-diameter is +2′′·11, which is very nearly the same as that determined in the year 1861 with the Meridian Circle.

For the Moon, a correction is applied if necessary to the transit of that limb which is defective on account of imperfect illumination (computed in the usual way), and the difference of the transits of the two limbs is then compared with that found by doubling the value given in the Nautical Almanac in the Section of Moon-Culminating Stars, corrected if necessary for difference of longitude. The mean of two errors of the tabular duration of passage of Semi-diameter is -0055. No observations of N. P. D. of both limbs were made in 1862 for the determination of the vertical diameter. The comparisons of the observed and tabular values of the diameters of the other planets need no explanation, excepting that those for Mercury, Venus, and Mars, have been corrected for defect of illumination. The application of the tabular errors of the diameters of the planet

Venus to determine the correction to be applied in 1862 has been already given (p. xxvii).

Right Ascensions and North Polar Distances.-The Mean Solar Times are computed by means of the Table given in Warnstorff's edition of Schumacher's Hülfstafeln. The sidereal time at mean noon given in the Nautical Almanac is increased by 083 for the assumed West longitude of the Radcliffe Observatory, namely 5m 28-6; the difference between this and the right ascension of any object is taken, and converted into the corresponding interval of mean solar time by the numbers in Warnstorff's Table. For cases wherein the N. P. D. of an object has been observed without a corresponding observation of R.A., the right ascension used for computing the mean solar time is that given in the Nautical Almanac, corrected, if it appears necessary, by means of the neighbouring observations.

The right ascensions of the Sun's centre are generally the means. of those deduced from the observations of the two limbs; if however one limb only has been observed, the duration of transit of semidiameter is taken without alteration from the Nautical Almanac, as those observed are not available for correction on account of the large values given by Mr. Lucas. It is also noticeable that the tabular errors of the right ascensions as observed by Mr. Lucas are too large. Now, on referring to the volumes of the Radcliffe Observations for the years 1840, 1841, and 1842, at which time Mr. Lucas observed with the Transit Instrument, it is found that no such peculiarity as that just pointed out existed in 1840 and 1841, but that in 1842 the tabular errors of right ascension are quite as large as in the observations under discussion. As the greater portion of the transits of the Sun were taken by Mr. Lucas the series cannot be considered to be available for the correction of the assumed equinox. The tabular R. A.'s are taken from the Nautical Almanac, corrected for longitude.

In general both limbs of the Sun have been observed in N. P. D., and the mean of the N.P.D.'s, corrected for discordance of direct and reflexion observations, &c., is taken for the true observed N.P.D. When one limb only has been observed, the semi-diameter of the Nautical Almanac is applied without correction.

The observed R. A.'s of the Moon's centre are derived from the observations of the limb by application of the Nautical Almanac "duration of passage of semi-diameter." The tabular R.A. is found by applying to the R.A. of the limb given in the Section of MoonCulminating Stars in the Nautical Almanac, corrected for longitude, the duration of passage of semi-diameter given in the same Section, similarly corrected if necessary.