PLACE OF A PLANET Sec. 



N". 11 r. 



To determine the true Place of a Planet, in an Elliptical Orbit, 

 direBly from the mean Anomaly, by Converging Ser es, by 

 David Rittenhouse, L. L. D. Prcfidcnt A. P. S. 



Read Feb.T ET X := the ccceiitricity, y the mean anomaly in 



5.1796- _£ J the arch of a circle the radius whereof is i. And 



a, an arch required. 



For the upper half of the orbit, let ;-j7^, ^= n, and jt7» ^= 2. 



Then a ■=. z ■\- - z^ -\- s + -— s:' + 



6 12 120 18 90 S'^40 



55»' 11;;' 4I;;« » _ ^^ ^,^_ 



1296 S64 60480 362880 



Find the log. of the natural cofme of a, and the log. of the 

 fame cofme + x, and add the difference of thefe two logarithms, 

 and likewife the complement of the log. of the conj. femidiame- 

 ter, and the log. cotang. of a, together, the fum will be the 

 log. cotang. of the true anomaly. 



For the lower half of the Orbit. 



Let y, be the mean anomaly from the lower apfis, -3-. zz m, 



and -^ = 2. 



Then a = z — 'L%^ + JL+^z^ — '^ + "l + J!^z' 



6 12 120 10 90 5040 



+ CC«* IIh' 4 Ik;/ » on 

 ■'^ 4, L _I I z &C. 

 1296 ^ 864 ^ 60480 ^ 362880 



Take the difference between the log. of the nat. cofine of a, 

 and the log. of the fame cofme — x, and fubtraft this diff. — the 

 comp. above mentioned from the log. cotang. of<2, the remainder is 

 the log. cotang. of the true anomaly, counted from the lower apfis. 



If the co-efficients prefixed to the powers of z, be computed for 

 any particular orbit, and their logarithms ufed inflead of the 

 numbers themfelves, the calculation will afterwards be very 

 fimple for any degree of mean anomaly in that orbit, as will ap- 

 pear by the following example. 



In the very elaborate tables of Mr. Zach, publiflied in 1792 



the eccentricity of the Earth's orbit is affumed .0167923, confe- 



quently log. of the lefler femidiameter will be — 1.9999387, its 



D complement 



