determining the Orbits of Comets. 127 



hilated in Olbers's method, likewise takes place, and conse- 

 quently the problem is nearly rigorously solved ; for intervals 

 not quite equal the greater part of their amount at least is de- 

 stroyed. These series show, likewise, the reason of Dr. Ol- 

 bers's remark at the end of the preceding paper, that it is gene- 

 rally somewhat more accurate to take 



[Vr"] [R/ R"] 



[r r'f - [R R'] 



instead of the more simple ratio of the intervals of time. The 



error is less as long as r Z. fy % but when r7^2 the ratio 



of time is to be preferred. 



From the explanation here given the great success which 

 has attended Olbers's method on almost every occasion may 

 be sufficiently accounted for. The quantities of the first 

 order are in every case completely taken into account. In the 

 first place the possibility of the case of equal intervals, not a 

 very rare one, in which even the terms of the second order 

 with respect to the intervals are perfectly destroyed, and a ri- 

 gorous solution is obtained ; and in the next place the cer- 

 tainty that even in the cases of unequal distribution of the time 

 between the observations only a very small part of these latter 

 terms is neglected, insure to this method the preeminence 

 over all the other known methods, which, as it is founded in 

 theory, has from its very first publication been confirmed by 

 practice. 



Should it be apprehended that in cases of very unequal in- 

 tervals the error of the approximate supposition might too 

 much affect the result, this supposition may be amended as 

 soon as the calculation has been carried on to the end of the 

 trials. The values of r, r" which are known, are generally 

 correct to quantities of the second order. Deducing from 

 them the value of r 1 or making 



r'=i(r"+r)-i T -^(r JI -r) 



and likewise 



dS _ r»-r 

 dt~ t' 

 we shall obtain correctly to quantities of the third order, 

 ( o^ [r>r"1_ t r 4 (t-tV ,, t»-r \ 



If we calculate besides accurately from the existing data 



[R' R"l 



>p pn an( * designate the approximate value of p brought out 



