VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 185 



2 2 2 



-r~r 



SI + ^-"H- lIx/ y^t ^ 9^+» + 3'x/ X ^ + 9X 25/ -« + 51x7 X y _j. gjj._ 

 1 ^ <* 5I <* tTj '" ' 



and the area -J-m X h, will be the sector, as before. 



For the point s being on the contrary side of the centre, to what it was be- 

 fore, it will easily appear, that the change of -\-f into — f, must reduce one 

 case to the other, without any other proof. 



Carol. — Hence, if the number n be taken equal to ^Z —r-, or in this case y'^, 

 then the series for the sector will want the second term, as in the former it 

 wanted the third. 



Definition. — The angle called by Kepler the anomalia eccentri, is a fictitious 

 angle in the elliptic orbit of a planet, being analogous to the area described by 

 a line from the centre of the orbit, and revolving with the planet from the line 

 of apsides; in like manner as the mean anomaly is a fictitious angle, analogous 

 to the area described by a line from the focus. 



Otherwise, if c be the centre, s the focus of an elliptic orbit described on 

 the transverse axis ap, and the area nsp in the circle be taken in proportion to 

 the whole, as the area described in the ellipsis about the focus, to the whole: 

 then is the arch of the circle pn, or the angle nop, that which Kepler calls the 

 anomalia eccehtri. 



This angle may be measured either from the aphelion, or from the perihe- 

 lion: in the following proposition it is supposed to be taken from the peri- 

 helion. 



Proposition 1. — The mean anomaly of a comet or planet, revolving in a given 

 elliptic orbit, being given; to find the anomalia eccentri. — The solution of this 

 problem requires two different rules; the first and principal one serves to make 

 a beginning for a further approximation, and the other is for the progression 

 in approximating nearer and nearer ad libitum. 



1. The rule for the first assumption : let ^,/, and j&, stand, as before, for the 

 semi-transverse axis of the ellipsis, the semi-distance of the foci, and the peri- 

 helion distance ; then taking the number n equal to \/ 5 + ^25 + ^j let t stand 



for ■ — ; and p for—- — ; — , or^T; which constant numbers, be- 



nnt — nn — l.p nnt — nn—\.p t ' 



ing once computed for the given orbit, will Serve to find the angle required 

 nearly by the following rule. 



Let M be the number cf degrees in the angle of mean anomaly to the given 

 time, reckoned from or to the perihelion ; and supposing r, as before, to stand 

 for 57,2937 &c. degrees; take the number n = y'— m, and let A be the angle 



VOL. VIIT. 1 atlJ o B B 



