Eambaut — Shape of Earth's Shadow on the Moon's Disc. 53 

 Then a = s - p, 



R 



and <v/(« - If + {(5 - r,) 2 = 



sin a 



it „ <. y R cos jR sin 



therefore a - £ = — : , and B - n = : . 



sin a sin a 



Also if £ denotes the zenith distance of the moon, we have 



£ = - R cos £, and tj = ii! sin X,. 

 Therefore 



-B 2 i2 2 7i! 2 



a = ^T~ ( Sin ' ~ Sin2 ^> b = "^T- ( COs2 # - ^n 2 <r), C = -r-r- (COS 2 ff), 



sin'ir snr<x y sm 2 <x 



i2 2 



A = . „ (sin cosfl), 

 sin 2 a 



i2 3 



J = -r-r- [(sin 2 - sinV) cos Z, - sin cos sin $ + cos 9 sin <xl, 

 sm 2 o- u J ' 



-B 3 



w = -T-7- [ sm cos cos £ - (cos 2 - sin 2 a) sin £ - sin sin <rl, 



sin V L y J ' 



R i 



d = -r-v - f(sin 2 - sinV) cos 2 £ + (cos 2 - sin 2 <r) sin 2 £ 

 sinV LV v ; 



- 2 sin 9 cos sin £ cos X, + 2 sin a cos cos £ 



+ 2 sin a sin 9 sin £ - 1]. 



A1 R(l -sinsTCOs£) . i^sinp 



Also u = — — — ; -, and r = — : — - . 



sm zs sin zs 



Hence 



, i^(l-sinwcos^) , r . . 1 



/ = : -, and d = R 2 (1 - snm cos £) 2 - sm 2 p] -^ . 



smzs L ^ ' rj surra 



If we take the moon on the horizon, in which case the effect of 

 parallax will be greatest, we have 



cos £ = 0, and sin £ = 1. 



Also, if 180°- <j> be the angle between the directions of the sun 

 and moon as seen from the earth's centre ; then 



9 = (p + zs. 



Now is a small angle, at its greatest about equal to vs. Hence 

 9 can never exceed 2°. "We may therefore put 



sin 9 = <p + vs, and cos 9 = 1. 



