VOL. XL.] PHILOSOPHICAL TRANSACTIONS. Iff 



If the eccentricity be less than .V, then take the number n equal to 

 v/^ and — X - will be nearly equal to the sine of the submultiple part of the 

 anomalia eccentri, denominated by the number n, as before. 



It is needless to observe, that the like rules would obtain in hyperbolic orbits, 

 mutatis mutandis. But that which perhaps may not appear unworthy of being 

 remarked, concerning this sort of solution from the cubic root, is, that 

 though the rule be altogether impossible, on a total change of the figure of 

 the orbit, either into a circle, or into a parabola; yet it will operate so much 

 better, and stand in need of less correction, according as the figure advances 

 nearer, in its change, towards either of those two forms. 



That the use of the method may better appear, it may not be amiss to add a 

 few examples. 



The following are two for the orbits of planets, one the most, and the other 

 the least eccentric; but which are more to show the extent of the rule, than to 

 recommend the use of it in such cases; for there are many other much better, 

 and more expeditious methods, in orbits of small eccentricity. The other two 

 examples are adapted to the orbits of two comets, whose periods have been 

 already discovered by Dr. Halley; the one is to show the use of one of the 

 rules in the first corollary, and the other is to explain the use of the other rule. 



Example 1 . — For the Orbit of Mercury. — If an unit be put for the semi- 

 transverse axis t, the eccentricity 0,20589 ^'^^ become f, and the perihelion 

 distance J!) will be 0,79^11 ; therefore by means of the number b, given as be- 

 fore, the constant numbers for this orbit will appear to be, n = 3,56755, 

 T = 0,5857271, p = Y T = 0,4651319, and hence — 5Z_ = 0,0085965. 



Example. — Suppose m, the mean anomaly from the perihelion, to be 120" 

 00' 00", to which it is required to find the anomalia eccentri. 



Here, since the mean anomaly m is not many times more than the limiting 

 angle -^R-v/p, which in this orbit is about 74°, recourse must be had to the 



general rule in the proposition. 



3x 

 The number n then, which is (/— m, will be = 1,0104195 ; which, found, 



gives nV i -f \/i + ^ = 1,0389090; and also N\/i - ^/i + -^= - 

 0,4477126. Therefore the sum of both, under their proper signs, viz. 

 0,591 1964, will be the sine whose arch 36°,24195 is the angle A; the multiple 

 of which n X A = 129°,295503, will be the angle to be first assumed for the 

 anomalia eccentri. 



For a further correction ; this angle, now called b, whose sine is suppose y, 



BB 2 



