Ixxii ASTRONOMY. 



i. Differential formulas. 



The general differential relations for the altitude and azimuth and the declina- 

 tion and hour angle systems of coordinates are : — 



</;; = — cos q d^ -\- sin g cos h dt -{- cos A deft, 

 sin z dA = sin g d8 -\- cos g cos S d/ — cos z sin A d<f>. 



d8 = — cos g dz -\- sin g sin z dA -\- cos / d(f), 

 cos 8 dt =z sin g dz -\- cos g sin 2 ^/^ -|- sin S sin / dtft. 



(0 



(2) 



The following values derived from (i) are of interest as showing the dependence 

 of z and ^ on / in special cases : — 



Xdt) ydf) 



For a star in the meridian = o, 



cos 8 



sm z 

 For a star in the prime vertical = cos ^, = sin ^, 



For a star at elongation = cos 8, = o. 



3. Relations of Different Kinds of Time used in Astronomy. 



a. The sidereal and solar days. 



The sidereal day is the interval between two successive transits of the vernal 

 equinox over the same meridian. The sidereal time at any instant is the hour 

 angle of the vernal equinox reckoned from the meridian towards the west from o 

 to 24 hours. 'The sidereal time at any place is o when the vernal equinox is in 

 the meridian of that place. 



The solar day is the interval between two successive transits of the sun across 

 any meridian ; and the solar time at any instant is the hour angle of the sun at 

 that instant. The solar day begins at any place when the sun is in the meridian 

 of that place. 



The mean solar day is the interval between two successive transits over the 

 same meridian of a fictitious sun, called the mean sun, which is assumed to move 

 uniformly in the equator at such a rate that it returns to the vernal equinox at 

 the same instant with the actual sun. 



Time reckoned with respect to the actual sun is called apparent time, while 

 that reckoned with respect to the mean sun is called mean time. The difference 

 between apparent and mean time, which amounts at most to about 16"', is called 

 the equation of time. This quantity is given for every day in the year in 

 ephemerides. 



The sidereal time when a star or other object crosses the meridian is called the 

 right ascension of the object. The right ascension of the mean sun is also called 

 the sidereal time of mean noon. This time is given for every day in the year in 

 ephemerides for particular meridians, and can be found for any meridian by allow- 

 ing for the difference in longitude. 



The time to which ephemerides and most astronomical calculations are referred 



