l(5Q PHILOSOPHICAL TRANSACTIONS. [aNNO I764. 



of any place ; for, the tangent ot then becoming infinite, the angle PTa 

 vanishes ; therefore the mean equinox a will come to the meridian at the same 

 instant of absolute time, as if the pole had not been translated from ? to a ; 

 and the difference of time between the sun s coming to the meridian, and a 

 fictitious sun u, supposed to move uniformly in the equator, with a motion 

 equal to the sun's mean motion in longitude, or the equation of time will be 

 therefore measured by aqs ot apu, the difference of their right ascensions 

 reckoned from the same point a. It will also, by the like reasoning, be mea- 

 sured by Bas CO bpu, the difference of their right ascensions reckoned from the 

 same point b ; for b being the equinox, when the pole is at a, the absolute time 

 of the point b passing the meridian of any place will remain the same as if the 

 pole had continued at p ; whence the proposition easily follows, in like manner 

 as above. 



It may be now proper to show how the equation of time, as affected by the 

 nutation of the earth's axis, ought to be computed. This may be done two 

 ways. The " first follows from what has been just laid down : correct the mean 

 right ascension of the sun upa, by the precession of the equinoxes in right 

 ascension apb (which is always to the precession in longitude ba, as cosine of 

 the obliquity of the ecliptic, to the radius, or as 12 to 13 nearly) the difference 

 of the sun's mean right ascension thus corrected bpu, and the sun's apparent 

 ascension bqs, turned into time, is the true equation of time." 



Otherwise, the effect of the nutation of the earth's axis on the equation of 

 time, if thought deserving notice, for it can never exceed ^ of a second of time, 

 might be computed from the angle pTa = pa X sine of tpd -f- tq, which, 

 supposing the nutation of the pole to be performed in a circle, whose radius is 

 8*, or a mean between the two conjugate semi-axes of the ellipsis, in which it 

 really moves, is = 8" X tangent of the sun's declination X cosine of the dif- 

 ference of sun's right ascension and the longitude of the moon's ascending node. 

 But this is not the only mistake in the computation of the equation of time 

 in the Connoissance des Mouvements Celestes, though it may exceed one 

 second of time. M. Delalande says that the sum of the equation of the sun's 

 centre, the difference between his longitude and right ascension, and the sum 

 of the 4 little equations, must be converted into mean solar time, in order to 

 find the equation of time ; and adds, that no exact equation table could be had, 

 before this time, for 3 reasons, one of which is, that it has always been the 

 practice to convert the equation of the sun's centre and the difference between 

 his longitude and right ascension into time of the primum mobile, instead of 

 mean solar time, which, says he, may produce an error of 24- seconds. 



Now I must here freely own, that as I could not, without some reluctance, 

 and only from the fullest proof, allow all the mathematicians and astronomers. 



