APPENDIX. 291 



cos a = sin 8 sin i 



cos b = cos 8 sin i cos cos i sin e 



cos c = cos 8 sin i sin e -\- cos i cos 



sin 2 a -f- sin 2 -f- sin 2 c = 2 



cos 2 a -f- cos 2 b -f- cos 2 e = 1 

 cos ( A B) = cotan a cotan b 

 cos (B G} = cotan b cotan c 

 cos (A C) = cotan a cotan c. 



58. 



If in the formulas of article 56 of the Appendix, the ecliptic is adopted as 

 the fundamental plane, in which case e = ; and if we put 



n = long, of the perihelion 

 sin a = k x A = K C (n 8 ) 

 sii\b = k y B = K y (n 8) 

 sine Jc z C=K Z (n 8) 

 we shall have 



k x sin (K x (n Q )) = cos 8 



Jc x cos (K x (n 8 )) = sin 8 cos i 

 Jc x sin K x = cos 8 cos (n 8 ) sin 8 sin (it 8 ) cos f 

 z cos K x = [cos 8 sin (it Q, ) -|- sin Q cos (JT 8 ) cosz] 



which can easily be reduced to the form, 



& x sin K x = cos 2 k i cos TT -J- sin 2 J z cos (TT 2 8 ) 

 # z cos K x = [cos 2 J z sin n -j- sin 2 ^ ism (it 2 8 )] 



and in like manner we should find 



k s sin K y = cos 2 i z sin n sin 2 i z sin (it 2 8 ) 

 = cos 2 J / cos it sin 2 J i cos (TT 2 8 ) 

 # z sin _ff^ = sin i sin (TT 8 ) 

 k z cos ^ = sin i sin (JT 8 ) 



