ASTRONOMY. 141 



Also, tan. NM'O = ^^, so that the angle NM'O 

 = S'NM' is given ; let this angle = ft 



then NM' = t J (n + m) + A*= r t, 



y 2 2 

 (n + ni) -f- A. Hence, if SM = fl, 



S'M' =. rt 2 art X cos/s+ 5 



and if the sum of the semidiameter of the moon 

 and shadow be called s 9 we have the equation 

 r* t* 2 a r t cos /3 + a* = s% for determining/. 



TT a + V st a* sm/3* 



Hence = ' > or 



r r 



a -f- V s* a* sin /s* 

 r 



These two values of t give the time of the beginning 

 and end of the eclipse. 



The least distance of the centres, is found by substi- 



tuting - for t in the formula 



S'M' = V r*t* 2 a r t X cos * + a*, 



S'M' 



