VOL. LIV.] PHILOSOPHICAL TRANSACTIONS. l®^ 



meridian ps, at s, then the spherical angle spd is the distance of the sun from 

 the meridian pd, when the pole is at p, and sqd is his distance from the same 

 meridian, when the pole is translated to a. Let pt, aT, meeting in t, be tan- 

 gents of the meridians ps, qs, in p and a ; tqd being the external angle of the 

 rectilineal triangle tps, the angle PTa is = tqd — tpd := saD — spd, and 

 therefore is a measure of the alteration of the time of any meridian of the 

 earth's coming to the sun at s, produced by the translation of the pole from p to 

 a. Now the sine of ptq is to the sine of tpq, as pa to tq ; whence, calling 

 the radius unity, and taking pq, on account of its smallness, = the sine of pa, 

 and the anorle PTa = the sine of PTa, we have PTa = ^*» x "°^ ^f Q _ ^.j^^ 

 translation of the pole X the sine of the right ascension of the sun or star 

 reckoned from the meridian in which the pole moves, divided by the tangent of 

 the polar distance, or, which is the same thing, multiplied by the tangent of 

 the declination. Therefore, as pa, arising from the nutation of the earth's 

 axis, never exceeds g^", the greatest value of PTa, for the sun can never ex- 

 ceed Qi" X tangent of 234-° the sun's greatest declination, = 4".l, which 

 answers to about -^ of a second of time : and so much, and no more, may the 

 sun come sooner or later to the meridian, on account of the nutation of the 

 earth's axis: whereas, if the equation of the equinoxes was to be applied directly 

 in the computation, according to M. Delalande's methotl, it would sometimes, 

 namely when at its maximum of 18", produce nearly H second of time. 



But, though this demonstration may be admitted to be just, yet it may per- 

 haps be asked, wherein lay the fault of the method of computation here cen- 

 sured, and whether the time of the sun's coming to the meridian is not regulated 

 by his right ascension ? It may also be thought requisite that the true manner 

 of computing the equation of time, from the sun's right ascension, should be 

 shown. First, let it be observed, that when the pole is at p, a is the equinoc- 

 tial ]X)int, and when the pole is translated to a, some other point b is the equi- 

 noctial point : therefore the sun's mean right ascension upa is reckoned from a 

 and his apparent right ascension sas, computed from his longitude, corrected 

 by the equation of the equinoxes ab, or bs, is reckoned from another point b. 

 Now the equation of time is proportional to the difference between the sun's 

 mean and true right ascension, both reckoned from the same point ; so that if 

 the sun's mean right ascension be reckoned from a, his apparent right ascen- 

 sion, in this case, should be reckoned from a too ; or if the apparent right 

 ascension be reckoned, more properly, from the apparent right equinox b, his 

 mean right ascension, for this purpose, should be reckoned from b also. For it 

 is plain, from what has been said above, that no small motion of the pole p can 

 at all affect the absolute time of a star in the equator's coming to the meridian 



