130 DETERMINATION OF ELLIPTIC ORBITS. 



the determination of the orbit. This will be the case when we wish 

 simultaneously to correct the formula for its theoretical imperfection, 

 and to correct the observations by comparison with others not too 

 remote. The rough approximation to the orbit given by the un- 

 corrected formula may be sufficient for this purpose. In fact, for 

 observations separated by very small intervals, the imperfection of 

 the uncorrected formula will be likely to affect the orbit less than 

 the errors of the observations. 



The computer may prefer to determine the orbit from the first and 

 third heliocentric positions with their times. This process, which has 

 certain advantages, is perhaps a little longer than that here given, and 

 does not lend itself quite so readily to successive improvements of the 

 hypothesis. When it is desired to derive an improved hypothesis 

 from an orbit thus determined, the formulae in XII of the summary 

 may be used. 



SUMMARY OF FORMULAE 



WITH DIRECTIONS FOR USE. 

 (For the case in which an approximate orbit is known in advance, see XII. } 



I. 



Preliminary computations relating to the intervals of time. 



t l} t 2 , t z = times of the observations in days. 

 log & = 8-2355814 (after Gauss) 



A _32 A _ 



~ --~ 



7? -1 13 3 fl _ 113 3 _ 1 13 



12 12 12 



For control : A^ + B^+A 3 B 3 = Jr^. 



II. 



Preliminary computations relating to the first observation. 



X lt Fp Z l (components of ^) = the heliocentric coordinates of the 



earth, increased by the geocentric 

 coordinates of the observatory. 



i 9i> 1 (components of 3i) = the direction-cosines of the observed 



position, corrected for the aber- 

 ration of the fixed stars. 



