126 RELATIONS BETWEEN SEVERAL [BOOK L 



r9 ,-, ^ _ _ sin g sin 2 cu _ 



~ cos 2 2 o&amp;gt; sin (fg) sin (/-f 0) + sin 2 2 &amp;lt;a cosy 



The ambiguity here remaining is easily decided by means of equation 7, which 

 shows, that G must be taken between and 180, or between 180 and 360, 

 as the numerator in these two formulas is positive or negative. 



By combining equation 3 with these, which flow at once from equation II. 

 article 8, 



1 1 2e . , . j-, 

 --- r = sin/ smF 

 r r p 



1 1 2 . 2e , 7, 



- -4- -v = -- -- cos / cos F, 



r / p p 



the following will be derived without trouble, 

 [25] tan.F=^ ^7^1 ^ 



L J 2cosyyrr (r-(-r)cos/ 



from which, the angle cu being introduced, results 



[26] tanfc sm/sin2 w 



I- - 1 



cos 2 2 ojsin(/ g) sin| (f-\-g) sin 2 2 cocos/ 



The uncertainty here is removed in the same manner as before. As soon as 

 the angles F and G shall have been found, we shall have v = F /, v = F-\-f, 

 whence the position of the perihelion will be known; also E= G g, E = G-\-g. 

 Finally the mean motion in the time t will be 



kt 



= 2,ff 2 ecosGsmff, 



a* 



the agreement of which expressions will serve to confirm the calculation ; also, 

 the epoch of the mean anomaly, corresponding to the middle time between the 

 two given times, will be G e sin G cos g, which can be transferred at pleasure 

 to any other time. It is somewhat more convenient to compute the mean 

 anomalies for the two given times by the formulas E e sin E, E e sin E , and 

 to make use of their difference for a proof of the calculation, by comparing it with 



It 



