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ON SOME APPLICATIONS OF THE METHOD 



quiring but one new logarithm. Table I., at the end of this article, contains numerical 

 values of 2 -^., with r 2 for the argument, supposing the interval to be one day. 



It is evident that this method cannot be advantageously employed, unless the inter- 

 mediate values of r between r and r„ are wanted. 



In the following example, r is the radius vector of Halley's Comet for each Green- 

 wich mean midnight, from August 1st to August 20th, 1835; using the elements given 

 in the Nautical Almanac for 1839. The influence of the second and higher differences 

 of F is insensible, so that J 2 n = F n+l , and the whole series is calculated directly from 

 Table I., without using logarithms after computing the constants r\, J 1 r 2 ', and *-?. 

 rl = 4.019789 ; z/ 1 r 2 = —0.055698 ; — *£ = —0.000032 7. 



!-? is taken from Table I., with the argument r 2 , the value of r 2 , next to r 2 , being 

 rl +l = r* + z/„'_, + F n . The correctness of the series maybe tested by computing r* 

 for August 20.5 by the usual methods. 



(13) The substitution of 



Tr 2 rf.r 2 2evA< 



V r = —r- = r sin. v 



2 e iv p a sin. m 



in equations (10) or (11) affords an accurate value of J' r„ when Fr] and its differences 

 are known. Or it may be obtained from the expressions, 



2e 



n — r„=: —r,r sin. M«i + J>o) sm - £ ("i — v o) = 2aesin. J («, + «,,) sin. £ (w, — « ), 



(14) '* '"~ p 



^ r o = (n+n)(n— »■„), 



e and p being the eccentricity and semi-parameter of the orbit, and w and v the eccentric 

 and true anomalies. 



