186 PHILOSOPHICAL TKANSACTIONS. [aNNO 1738. 



whose sine is n V ^+ \/j + $ + n V g "" V i + '^'' *^^^" ^^^ multiple 

 angle n X A, will be nearly equal to the anomalia eccentri. 



The truth of which will appear from the resolution of the cubic equation, in 

 the last corollary to the preceding proposition. 



Carol. ] . — If the quadruple of the quantity — ^ be many times greater or many 

 times less than unity; or, which amounts to the same, if the mean anomaly n, 

 be many times less, or many times greater, than the angle denoted by the given 

 quantity -^Ry^p, one or the other of which two cases most frequently happens 

 in orbits of very large eccentricity; then the theorem will be reduced to a sim- 

 pler form, near enough for use. 



Case 1 . — If M be many times less than 

 ^ Rv^p, then the angle a may be taken for that whose sine Is —. 



Case 2.— If M be many times greater than 

 — R\/p, then let a be the angle whose sine is n — -; and the multiple angle 

 n X A, according to its case, will be nearly equal to the angle required. ' 



Carol. 1. — In orbits of very large eccentricity, the perihelion distance p is 

 many times less than the semi-distance of the foci /, and the number 



n = \/5+ ^25 + ^; is always nearly equal to '/lO, or to the integer 3, either 



of which may be used for it, without any material error in the orbits of comets. 



2. The rule for a further correction ad libitum. 



Let M be the given mean anomaly, t the semi-transverse axis, as before; and 

 let B be equal to, or nearly equal to, the multiple angle n X A, before found; 

 then if p. be the mean anomaly, and x the planet's distance from the sun, com- 

 puted to the anomalia eccentri b ; the angle b taken equal to b -| — X m — /*, 

 will approach nearer to the true value of the angle sought; and by repetitions 

 of the same operation, the approximation may be carried on nearer and nearer, 

 ad libitum. 



This last rule being obvious, the explication of it may be omitted at present. 



Scholium. — In this solution, where the motion is reckoned from the peri- 

 helion, the rule is universal, and under no limitation. But had the motion 

 been taken from the aphelion, the problem must have been divided into two 

 cases: one is, when the eccentricity is less than -^x the other is, when it is 

 not less, but is either equal to, or more than in that proportion. 



If the eccentricity be not less than -rV> then the same rule will hold, as be- 

 fore, only putting the aphelion distance, suppose a instead of the perihelion 

 distance p, and substituting — / for -^f in the rule for the number n. 



