APPENDIX. 325 



VIII. 



tan U = tan i sin (t Q ) 

 tan If&quot; tan 6 cos (Z&quot; 7 Z ) 



rin(r-0 = tan i COB (f 8) 



i the inclination is always positive, and less than 90. The upper signs are to be 

 used when the motion is direct ; the lower when it is retrograde. 



IX. 



= tan (L - Q ), **&quot;-*) = tan ( &quot;- fi ). 



COS v 



COS 4 



&quot; 



L and L&quot; are the longitudes in orbit. 



SECOND CONTROL. 



The value of k before computed must be exactly 



k = y/ [V 2 + r &quot;* 2 / /&quot; cos (II&quot; )]. 



/ 



X. 



1 _ cos ^ (L n) 



cos (If &quot;L ) 



V// V 7 &quot; V ? 



TT, the longitude of the perihelion, is counted from a point in the orbit from which 

 the distance, in the direction of the order of the signs, to the ascending node, is 

 equal to the longitude of the ascending node. 



XL 



The true anomalies are 



v = L n, v &quot; = L &quot;n. 



With these the corresponding M and M &quot; are to be taken from BARKER S 

 Table, and we have then the time of perihelion passage 



T= f =F M q* n = f =F M &quot; q% n, 



