﻿OF MECHANICAL QUADRATURES. 193 



conditions are not susceptible of any precise statement, being affected by a variety of 

 circumstances. 



We now proceed to the application of the method of quadratures to some examples 

 in astronomy. 



II. Computation of the Radius Vector r in Orbits of any Eccentricity. 



The differential expression which seems best fitted for this purpose is 



Fr 2 = 2 ^ — *£. (12) 



JV being the second differential coefficient of r 2 , and ^ the reciprocal of the semi- 

 axis of the orbit, which is == in the parabola, and negative in the hyperbola ; fi is the 

 sun's mass + the mass of the planet, which, when x is the number of mean solar days 

 in each interval, and the mass of the body is neglected, may be expressed by ^, = k' 1 T, 

 log. k = 8.2355814. 



We shall suppose that r* and J l r] = r\ — r* are given, with the semi-axis of the 

 orbit, and that it is required to compute a series of values of r from * = to t = nx; 

 x being taken such that i£ is very small. 



The first step is to compute F and F t from (12) using a, r„, and r\. As 2 -£ is a 

 constant, each value of F requires only the calculation of ?£ to be repeated for each 

 interval. Neglecting, at first, iV d] in (8), which is of the order of the fourth differ- 

 ences of r 2 , F l added to J\ gives J\, and thence r\, very nearly; this value of r\ used 

 in (12) gives a very exact value of F„ because whatever error there may be in r 2 , it 

 is multiplied in F it by the coefficient p, which we may make as small as we please by 

 diminishing x. 



Arranging now F , F t , and F 2 in a vertical column, and taking the differences, 

 < = F r 2 



t=\ F, dl r\ J} 



t = 2 Ft r 2 



we obtain d], with which we correct the first value of r*. This correction, where 

 the intervals between r\, r\, &c, do not exceed two or three days, or as many de- 

 grees of heliocentric motion, will be exceedingly small, and wholly insensible on F„ 

 which may next be used to find r\ = r\ + J\ + F 2 ; it being seldom necessary to notice 

 the differences of F above the second. 



After F is known for two or three intervals, r 2 follows for the remaining intervals 

 without intervening approximation. In this manner r s may be computed with accuracy 

 in the seventh place of decimals, with logarithmic tables of five places, each value re- 



26 



