determining the Orbits of Comets. 21 



the linear velocity of the comet at the time of the second ob- 

 servation is = / ( — ) the linear velocity of the earth being 



assumed = 1, while the ellipticity of its orbit is neglected, so 

 that approximately 



consequently s 7 4. Both cases, comets very near to the earth 

 and the sun, and comets very distant from the sun, are rare, 

 especially the latter. Commonly q will have a positive value. 

 Hence it follows that employing first s , then the following 

 value 5,, next the one resulting from that one s^ and so on, we 

 shall obtain alternately values too small and too great, an ad- 

 vantage of no small consequence for the rapid approximation. 

 For the corrections, or that which is to be added to the as- 

 sumed value in order to obtain the true one, will be 

 d Sj = — q ds 

 d s 2 = — qds x = + q 9 d s 

 ds 3 = — q d s 2 = — q^d s , &c. 



We find indeed, in the four examples calculated by Olbers, 

 this change of signs occurring three different times. Only in 

 the case of the first comet of 1 805, where the case above re- 

 ferred to occurred, the successive values are s = 2, s x = 1*413, 

 s 2 = 1*328, s 3 = 1*318, which are all in excess. 



By means of the same consideration we may, when three 

 values have been calculated, approximate to the truth by an 

 easy interpolation. We will suppose s }9 s 2 , s s any three suc- 

 cessive values of s, each of which has been derived from the 

 one next preceding it. Let us find the arithmetical differences : 



s 2 ; j 2 s , 



S 3 



we shall have approximately 



A s x = ( 1 4- q) d s x 

 A h = — ( l + 9) 9*ds l 

 J%=_(1 +q)*ds l9 



consequently, 



ds x = 



ds 3 



(*hY 



J 2 5 2 



(Js,y 



and generally an approximate value of q, 



