determining the Orbits of Comets, 23 



are, of course, likewise here applicable. The smallest valueof 

 5 is, agreeably to what is said above, B + B"; when this value 

 is itself nearly = 2, in the cases of comets which are near 

 the quadrature, or much smaller than 2, for comets which are 

 nearer to the conjunction or opposition, a greater or less value 

 may be chosen in the beginning, especially with a due regard 

 to the values of b and 6" compared to those of c and c" . 



In order to give an application of these formulae I will give 

 here the calculation of the second comet of 1813, which has 

 become a model by Gauss's paper, and I will apply it to the 

 method of trials here explained. 



The observations of Gottinger of the 7th, 14th, and 21st of 

 April, 1813, gave for this comet the following data : 



t = 7*55002 



t' == 14*54694 



*" = 21-59931. 



8 == + 29° 2' 0" 

 8' = + 22 52 18 

 8" = + 9 53 12 

 log R = 0-00091 

 logR/ = 0-00175 

 log R" = 0-00260. 

 Hence will be found: 



log M = 9-75799 



G= 113° 43' 57" 

 logg = 9-38029 

 H = 109° 5' 49" 

 ? = 44 13 9 



, . log/* = 9-81477 



log A = 9-22527 

 logB = 9-98706 

 log B" = 9*86038 

 log b = 9-75645 

 log b" = 0-05028 



c = + 0-31365 



c" = + 0-95443. 



As a first trial (although it may be easily seen that s must be 

 considerably greater than 2) I assume 



$ = 2, logs = 0-30103. 

 We have next, since 



t" -t= 14-04929 



log 2 K (t" - t) = 9-68427, 

 whence the calculation stands thus : 



