SECT. 1.] TO POSITION IN THE ORBIT. 45 



[1], article 37 ; the second, the determination of E from A and B, which rna y be 

 done directly, either by means of the equation 



or by this, 



sin E= 



the third, the determination of v from E by means of equation VII., article 8. 

 The first operation, we will bring to an easy calculation free from vague trials ; 

 the second and third, we will really abridge into one, by inserting a new quantity 

 C in our table by which means we shall have no need of E, and at the same 

 time we shall obtain an elegant and convenient formula for the radius vector. 

 Each of these subjects we will follow out in its proper order. 



First, we will change&quot; the form of equation [1] so that the Barkerian table 

 may be used in the solution of it. For this purpose we will put 



5 5e 



j 



: 



from which comes 



ITK A OCA ? 



7 5 tan i?f 4- 2 5 tan %w s = 



denoting by a the constant 



If therefore B should be known, w could be immediately taken from the Barkerian 

 table containing the true anomaly to which a nswers the mean motion -^ ; A will 

 be deduced from w by means of the formula 



A = fi tan 2 i iv, 

 denoting the constant 



5 5 e , ,, 



r+^ by ^- 



Now, although B may be finally known from A by means of our auxiliary table, 

 nevertheless it can be foreseen, owing to its diifering so little from unity, that if 

 the divisor B were wholly neglected from the beginning, w and A would be 

 affected with a slight error only. Therefore, we will first determine roughly w 

 and A, putting 2? = 1 ; with the approximate value of A, we will find B in our 



