638 PHILOSOPHICAL TRANSACTIONS. [aNNO I762. 



the altitudes of the moon and star be not less than J0°, this correction will not 

 be necessary. It will not be proper to make the obseiTations, if the altitudes of 

 the star and moon are either of them less than 4° or 5^, on account of the variable- 

 ness of refraction near the horizon. 



Bule 2. To find how much the distance of the moon and a star is increased 

 or diminished, on account of the moon's parallax. 



Add together into one sum the logarithm-tangents of half the sum, and half the 

 difference of the zenith distances, and the cotangent of half the distance of the 

 moon and star, all corrected for refraction ; the sum, abating 20 from the index, 

 is the tangent of arc the 3d, for which arc the 2d, found by the first rule, may 

 be taken, without any sensible error. 



Then if the zenith distance of the moon is greater than that of the star, take 

 the sum of this arc and half the distance of the moon and star ; but if the zenith 

 distance of the moon is less than that of the star, take the difference of the said 

 arcs ; the tangent of the sum or difference, which may be called the parallactic 

 arc, added to the cosine of the moon's zenith distance, and the logarathm of the 

 moon's horizontal parallax in minutes abating 20 from the index, is the logarithm 

 of the number of minutes required, by which the apparent distance of the moon 

 from the star is always augmented by parallax, unless the zenith distance of the 

 star be greater than that of the moon, and at the same time, arc the 3d be greater 

 than half the distance of the moon and star, in which case the apparent distance 

 of the moon and star is diminished by the parallax. 



Therefore the number of minutes found by this rule is always to be subtracted 

 from the observed distance of the moon and star, first corrected for refraction, in 

 order to find the true distance, cleared from th^ effect of parallax likewise ; ex- 

 cept in the case specified, when the zenith distance of the star is greater than that 

 of the moon, and arc the 3d is at the same time greater than half the distance of 

 the moon and star, when the correction is to be added. In computing these cor- 

 rections, 4 places of figures beside the index will be sufficient. 



It remains to be found by calculation, at what hour under a known meridian, 

 the distance of the moon from the star will be the same as results from the obser- 

 vation, cleared of refraction and parallax. For this purpose it is necessary to 

 compute the moon's longitude and latitude, and horary motion both in longitude 

 *nd latitude, from the most exact tables, for the time under the known mendian, 

 which is judged to correspond nearly to the given time of observation under the 

 unknown meridian. The mean motions of the sun and moon he took from very 

 exact tables, which he received as a present from the ingenious Mr. Gael Morris, 

 composed by himself, from the comparison of a great number of Dr. Bradley's 

 observations; to which he applied the lunar equations, as they stand m Mr. 

 Mayer's printed tables. After finding the mean longitude of the star at the pre- 



