132 RELATIONS BETWEEN SEVERAL [BOOK I. 



with this value of , we have x = 0.0372213, which value requires no further cor 

 rection, since is not thereby changed. Afterwards is found % g = 11 7 25&quot;.40, 

 and hence in the same manner as in example I. 



k(F G)= 333 53&quot;.59 log P = log fi cos } 9 9.9700507 



826 6.38 log Q = logoff sin 9 . 9.8580552 



F= 115959.97 9 = 3741 34 / .27 



v = - 100 .03 9 = 75 23 8 .54 



v +1235959.97 log# ...... 0.0717096 



G = 4 52 12 .79 For proving the calculation. 



-172238.01 . . O . om o97 



E = -f-27 7 3.59 



The angle 9 in such eccentric orbits is computed a little more exactly by 

 formula 19*, which gives in our example 9 = 75 23 8&quot;.57; likewise the eccen 

 tricity e is determined with greater precision by the formula 



e = 12 sin 2 (45 9), 



than by e = sin 9 ; according to the former, e = 0.96764630. 



By formula 1, moreover, is found log b = 0.6576611, whence logp= 0.0595967, 

 log a = 1.2557255, and the logarithm of the perihelion distance 



log j^:= tog a (l):= log $ tea (45 --*?)== 9.7656496. 



It is usual to give the time of passage through the perihelion in place of the 

 epoch of the mean anomaly in orbits approaching so nearly the form of the 

 parabola ; the intervals between this time and the times corresponding to the 

 two given places can be determined from the known elements by the method 

 given in article 41, of which intervals the difference or sum (according as the 

 perihelion lies without or between the two given places), since it must agree with 

 the time t, will serve to prove the computation. The numbers of this third ex 

 ample were based upon the assumed elements in the example of articles 38, 43, 

 as indeed that very example had furnished our first place : the trifling differences 

 of the elements obtained here owe their origin to the limited accuracy of the 

 logarithmic and trigonometrical tables. 



