PREDICTION OF SOLAR ECLIPSES. 143 



x = cos d sin (a — a) 



y = sin cos o' — cos o cos o' cos {a — a) 



/!=(---') Vcos^-t:') (2) 



ç = // p cos ip sin t =17 p sin z sin v \ 



rj = n s in <p' cos o'—I7p cos <p' sin o cos t = I7 p s in z cos v \ (3) 



sin <p' sin + cos <p' cos d' cos t= cos z ) 



D sin <r=(x—£) ) 



(4) 



D cos a = (j/ — '/}) ) 



where 



D = r+ r (1 — p sin tz cos z) for the partial eclipse \ 



D — r — r'{\—psiii7:cosz') ,, ,, total eclipse [ (5) 



D = r\l —[> sin z cos z') — r „ ,, annular eclipse j 



and o is the angle between the declination circle of the sun and the 

 line joining the apparent places of the sun and the moon, and is 

 reckoned positive towards the positive axis of x on the plane of pro- 

 jection. 



For convenience in computation, the formulae (1) may be put 

 in the following form — 



x =(a — a') cos d ^/cos (a— a) 



> *m 3/ > >r ,1 ■ -i« s ^ 1 ° ■■> 



ll — io — o ) Vfo.s' [o—o ) + -p Sbll 1 s- X- 



J ' ' 2 COS o 



.(!)« 



Eqns. (4) are the fundamental equations of the theory of eclipses. 



3. The outline of the sliadow vpon the earth at a given time. 



This outline is the curve on the surface of the earth, from every 

 point of which a contact of the sun's and moon's limits may be 

 observed at the given time. Or, it is the intersection of the cone of 

 the shadow with the earth's surface. 



Let T = the given time, reckoned at the first meridian and let 



