determining the Orbits of Comets. 1 1 



we get from the last equation 



sin J y = sin £ V 2. 



Of the three roots of the cubic equation, one only fulfills the 

 two conditions, that 



sin £ y 21 sin 45° or Z J V2, 



and is positive, viz. the one for which has been taken £ 90°. 

 For the lower sign we have this form : 



/cosjjA _ /£Osir\ 8 ■ 6K* 

 W2 / V a/2 ) - 2* (r + r')** 



or cos £ y = sin \ a/ 2. 



Here sin ^ must be Z £ a/ 2, and at the same time 

 cos £ y 7 £ a/ 2. These two conditions are only fulfilled when 

 is between 90° and 135°. It follows from this that when 

 the equation 



6Kt . A 



-7- T7 = sin 0, 



2^ (r + r')* 



gives a value for which is £ 45°, for the same data only one 

 solution is possible, for which x) — v Z. 180°. But when 

 7 45°, there are two solutions, and 180° — 0, the latter of 

 which answers to t/ — v 7 180°. 



The value of may be thus written : 



sin * = w ; 



and as we have both from the value of sin £ y as well as from 

 that of cos^ y 



sin y = 2* sin ^ a/ cos § 0, 



and likewise k = (r + r 1 ) sin y, 



we have the following complete system of formulae : 



2K* 

 1 = 



(11) Sm0 = V8 



(r+r'r 

 3* 



3 sin 1 

 sin 



a = r. I a/ cos & 



where 



- 2K* 



log2K = 8-5366114. 



By means of these formulae, log p has been calculated in the 



C2 



