206 Olbers's Method of determining the Orbits of Comets. 



da,' = ■ - , . d (r 1 



cos 8 2 



d V = cos (a' — 0') cos T . d <r', 

 or the ratio of these errors 



cos V .da! - ■ tan (g r — 0') 

 dV " " " sinS 7 ' 



If the orbit of the comet accords so nearly with the observa- 

 tions that d <r' is in itself very small, the magnitude of this 

 ratio need not further be taken into consideration. But if 

 from some cause or other d <r' is not insensible, and if the 

 greatness of the ratio indicates a very unequal distribution of 

 the errors, then we might have a reasonable inducement to 

 correct the orbit on a principle different from that which is the 

 foundation of Olbers's method : this may generally be most 



p" 

 easily effected by two or three hypotheses for p or Y — . The 



principle of Olbers may, agreeably to Bessel, be most simply 

 expressed in this way : that the orbit, while it strictly passes 

 through the two extreme places, likewise accords with the 

 great circle passing through the middle places of the sun and 

 comet. 



The exact agreement of the J5 7 obtained by the last results 

 of the calculation with the one involved in the observations, 

 may always be attained by Olbers's method ; but this can only 

 be done with perfect accuracy if the rigorous expression of 

 (13) or the correction of (26) and (27) is applied: and, in- 

 versely, this agreement, if perhaps attained by the approxi- 

 mate value of M only, proves that the rigorous expression 

 would not much differ from the approximate one. It affords 

 a proof that all further correction would be superfluous. If, 

 however, the calculated value of 2? should not be found to 

 agree with the other, without any error of calculation having 

 crept in, Carlini has proposed, if the difference is so small as 

 to effect only the last decimals of tan 2' 9 to use an artifice si- 

 milar to the one explained by Gauss in his summary view of 

 the calculation of the orbits of planets. Without applying 

 the strict correction, let a new value of M be calculated by 

 means of a value of X which deviates from the one derived 

 from the observations by an equal quantity, but in a contrary 

 direction, as the one found from the first calculation of the 

 orbit, and the new orbit thence deduced will give the value 

 deduced from the observations accurately, or at least very 

 nearly so. 



[To be continued.] 



