188 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738, 



and its cosine z, gives, by a known rule, f + ^z = 1,1304, for x the planet's 



distance from the sun ; and by another known rule b — —y = 1 20°, 1 6568, for u. 

 the mean anomaly to the anomalia eccentri b. Therefore the correct angle 

 B, = B + ^ X M — |«, will be 129°,14846 = 129°8'54",5, erring, as will ap- 

 pear from a further correction, about -rV of a second. 



This angle, being thus determined, will give by the common methods, 

 137" 48' 33-t", for the true anomaly, or angle at the sun : the sine of the true 

 anomaly being in proportion to the sine of the anomalia eccentri, as the semi- 

 conjugate axis, to the planet's distance from the sun. So that the equation of 

 the centre in this example is 17° 48' 33^". 



Example 2. — For the Orbit of Venus. — Supposing, as before, the mean 

 distance t to be unity, and the eccentricity/ to be 0,0069855 ; the constant 

 numbers for this orbit will be, p = 0,99301 15 ; n — 6,41 16 ; t = 1,562134 ; 



I' = 0,1551217 ; ^ = 0,0127571 ; and the limiting angle, ^r\/t-, will ap- 

 pear to be about 303 degrees. 



Example. — Let m be 120° OO' OO", as in the former example. Then, since 

 the mean anomaly is, in this case, not many times less than the limiting angle, 

 the general rule must be used as before ; according to which the number n will 

 appear to be 1,152585; the sine of a will be 0,32179)7; the angle a, 

 18°,77I32; and the multiple re X A, or angle b, for the first assumption of 

 the anomalia eccentri, will be 120°,354l6. 



This angle b will give, by the method before explained, the angle b = 

 120°,34555, or 120° 2l' 44* fer^, for the anomalia eccentri correct ; the error 

 of which will appear, on examination, to be but a small part of a second. 



In this example, the true anomaly is 120° 4l' 25", 1 ; and consequently the 

 equation of the centre no more than 41' 25", 1. 



Example 3. — For the Orbit of the Comet of l682. — To know the mean 

 anomaly of this comet, to any given time, it is to be premised, that it was at 

 the perihelion in the year l682, on Sept 4, at 21'' 22"*, equated time to the 

 meridian of Greenwich, and makes its revolution about the sun, as Dr. Halley 

 has discovered, in 754- years. 



The perihelion distance p is, according to his determination, 0,0326085 parts 

 of the mean distance t. So that the constant numbers for the orbit will be, 

 n = 3,1676061 ; T = 0,2054272; p = 0,00669867 ; and the Hmiting angle, 

 -^R\/p, will be about I9', or -f of a degree. 



of 



In the orbits of comets, the rule for the first assumption of the anomalia 

 eccentri, is generally sufficient without correction. 



