determining the Orbits of Comets.. 125 



which are accurate to the third power of the times inclusively. 

 These values are true for every body of our system. In the par- 

 ticular case of the orbit of the earth, the term of the third order 

 becomes quite insensible, on account of the very small factor 



—z — , and belongs, in some measure, to a higher order. The 



term of the second order in the only quotient which is employed 



in Olbers's method, viz. rR prj T > becomes for t = t" or for 



equal intervals quite equal zero. As one may always ap- 

 proximate to this most favourable case of the intervals of the 

 observations, as nearly as the observations from which the se- 

 lection is made will permit, the most considerable part, at least, 

 of the influence of terms of the second order may be destroyed. 

 Putting t = / . t" we have 



[R' R"] r_ f _ . l^jj ** \ 



R'] ~t" V */+l ' R' 3 * ' J 



where the factor , | , has for / =r J the value of £, for / = 2 



[R 



I- 1 



/+1 



the value ^, and for / = 3 the value ^ ; in the case, therefore, 

 of the very unfavourable distribution of the time in the ratio 

 of 1 to 3, one half of the influence of the terms of the second 

 order is annihilated. 



For the comet we may find an exact expression of the ratio 

 of the sector to the triangle by means of the auxiliary quantities 

 which have been used for the solution of Lambert's equation, 

 and these expressions will prove that here likewise in all cases 

 that can occur in practice, the terms of the second order are 

 so great in comparison of those of higher orders, that a solu- 

 tion satisfying the former is sufficiently accurate. The equa- 

 tion (6) may, agreeably to its import, be written thus: 



F = t» V2q = [r O + I [r H] —£^ 

 or substituting from (4) and (5) 



But as k — (r + r) sin y, 



we shall have, by substituting this value in (3), 

 m = (cos £ y + sin £ y) V (r + r 1 ) 

 n = (cos \ y — sin \ y) V (r + r^), 

 consequently : 



m — n = 2 sin \ y V (r + r') 



m n =s cos y (r -f r'), 



