•(14) 



|48 SHIN HIRAYAMA ; 



cos tp' sint= sinv 

 cos ip' cos t=—cos v sin o' 

 sin ip' = cos v cos d' 

 L = t-T +e 



We may easily find the geographical latitude from the geocentric. 



The sun is rising or setting at the given time at the places thus 

 determined, according as t, the hour angle of the sun is between 180° 

 and 360° or 0° and 180°. The eclipse is beginning or ending accord- 

 ing as cos (a—N) is negative or positive. 



7. To find the beginning and the end of the eclipse generally. 



In order to apply the preceding method of determining the rising 

 and setting limits, it is necessary first to find the extreme times be- 

 tween which the time t is to be assumed, or those limits of t between 

 which the solution is possible. The two solutions given by (13) 

 must reduce to a single one when the surface of the cone of 

 shadow has but a single point in common with the earth's surface. 

 Now, the two solutions reduce to one only when ^ = 0, and both 

 values of v become=ilf ; but if ^ = 0, then the numerator of the value 



of sin fr-A must also be zero ; and hence the points of contact are 



determined by the conditions 



D+m— 77/7=0; D-m+np=Q; 



or m = II (> — D ; m — D+ Up. 



There may be four cases of contact, two exterior and two interior. 

 The two exterior contacts are the first and last, or the beginning and 

 the end of the eclipse generally. The first interior contact corresponds 

 to the last point on the earth's surface where the eclipse ends at 

 sunrise ; the second, to the first point when it begins at sunset. But 

 these interior contacts can occur only when the w r hole of the shadow 

 on the principal plane falls on the earth. 



