determining the Orbits of Comets. 25 



or log s = 0-42616. 



A calculation made with this value would have given the 

 required value with perfect accuracy. 



As soon as s and consequently u 9 p and p" are found, the 

 elements of the parabola may be derived in various ways 

 from the two extreme observations. It will, however, be in- 

 teresting previously to investigate the degree of approxima- 

 tion obtained by the assumption in (15) and the applicability 

 of the formulas (16). 



[To be continued.] 



III. On the Summation of slowly converging and diverging 

 Infinite Series, By J. R. Young, Professor of Mathematics 

 in Belfast College, 



[Continued from vol. vi. p. 354.] 

 r |^HE foregoing examples have been chosen of the form 



S = b — ex + dx* — ex? + &c. 

 because it is only for slow series of this form that the trans- 

 formation furnished by the differential theorem offers any 

 practical facilities. When the series is of the form 



b + ex + dx 2 + ex + &c (D.) 



we may indeed convert it into 



1 f Ax A** 2 A 3 * 3 ' . T ,,_ 



i^i b - r^ + (i^y - (i _^ +&c - \ ••• ( E -> 



by simply substituting — x for x, in the formula (B.), and this 

 is a form to which the foregoing arithmetical process may be 

 applied. But it is easy to see that that process would con- 

 duct us to the original series, and would terminate in the ac- 

 tual summation of its successive terms ; for by the formula 

 (B.) it appears that the new series, into which (E.) would be 

 transformed by the process alluded to, would proceed accord- 



x / x \ 



ing to the powers of j j- f- hi)} instead of accord- 



x 

 ing to the powers of , as at present, that is, it would 



proceed according to the powers of x ; and, agreeably to this 



change, the factor ■ , which multiplies the series (E.), would 



become simply 1 ; so that (E.) would thus be converted into 

 a series of the form 



b + c'x + d'x* + e'x^ + &c. 

 Third Series. Vol. 7. No. 37. July 1835. E 



