Manner of estimating the Difference of Longitude inTime. 49 



one passage of the meridian and the next following be divided 

 into 24 hours. Therefore one step more is necessary for find- 

 ing the true difference of longitude ; namely, to change the arc 

 from hours of mean time, to hours at the rate of 24 to 360°, 

 which is done by dividing by r. Hence, if d be the difference 

 of longitude, we have, 



This is Mr. Henderson's formula in the Quarterly Journal. 

 If we suppose that a star is upon the first meridian at the 

 same instant with the moon, the star will be past the second 

 meridian when the moon arrives upon it; and the arc of the 

 equator between the star and the second meridian is evidently 

 the moon's variation in right- ascension, and equal to a in 

 sidereal time. The star separates from the first meridian 15° 



every hour of sidereal time, and — every hour of mean 



time; wherefore the whole arc of the equator between the star 



15° 

 and the first meridian, is equal to a x — in sidereal time. 



Hence the sidereal time of describing the arc between the 



two meridians is equal to 



15° 

 a X — — a : 



h . r 



and, as this time is at the rate of 1'' to 15°, we have 



15° 

 d = a X —, a. 



n . r 



This is Mr. Henderson's second formula, and it is precisely 

 the same with the first, since — = 15° 4- m. 



r 



The longitudes ascertain the relative positions of the terres- 

 trial meridians. If this end is to be accomplished by esti- 

 mations in time, it is requisite that the intervals elapsed be 

 proportional to the arcs of the e(]uatoi\ The difference of the 

 times of passing any two meridians must be the same pa"t of 

 an entire revolution that the intercepted arc of the equator is 

 of the whole circumference or 360^. If the time of a whole 

 revolution of the heavens be divided into 24 hours, there will 

 be the same number of hours in the difference of the longi- 

 tudes of two given meridians, whether the hours be long or 

 short, whether they be mean solar hours or sidereal hours. 

 The question relates entirely to different ways of measuring 

 the same quantities ; exact proportionality in the measures is 

 alone required ; their absolute magnitude is not considered. 



I cannot therefore subscribe to the decision ex cathedrdy 

 wliich appears in p. 121 of the Journal, namely, 



Vol. GG. No. 327. July 1825. G' '* It 



