VOL. XL.] PHILOSOPHICAL TRANSACTIONS. JSQ 



ThuSj suppose the mean anomaly m to be 0,072706, (as it was at the time of 

 an observation made at Greenwich on August 30, \682, at 7^ 42"" eq. t,) then 

 the general rule (which must be here used, since the angle of mean anomaly is 

 not above 4 or 5 times less than the limiting angle) will give n X a or b = 2° 

 12' 48", 7, erring about ^^ of a second from the true anomalia eccentri. 



But in these orbits, the rules in the first corollary to the second proposition 

 most frequently take place, especially the last ; and the calculation may also be 

 further abbreviated, by putting the square root of 10, or the integer 3, for the 

 number n. 



Example. — Suppose the mean anomaly to be 0°,006522, or 23",4792 : here, 

 since m is 50 times less than the limiting angle, the rule in the first case of 

 the first corollary may be used ; that is, to take the sine of the angle a = 



< X M 



np X R 



Therefore, if the number 3 be put for n, the sine of a, which is —.will be 



= 0,001 16367 ; and consequently the angle A will be 4'0",011 ; and the mul- 

 tiple angle n X a, to be assumed for the anomalia eccentri, will be 12'0",033, 

 the error of which will be found to be about ^ of a second. 



Example 4. — For the Orbit of the great Comet of the Year l680. — ^This 

 comet, according to Dr. Halley, performs its period in 575 years ; and was in 

 its perihelion on Dec. 7, 168O, at 23*' OQ"* eq. t. at London ; the perihelion 

 distance p is 0,000089301, in parts of the mean distance t: therefore sup- 

 posing the number n to be v'lO, the constant numbers for the orbit will be 

 T = 0,2000161 ; p = 0,000017862, and the limiting angle ^r^p, will be 

 about -^ of a second. 



Example. — Suppose the mean anomaly to be 3'31*,4478, or 0°,0587354], 

 (as it was at the time of the first observation made on it in Saxony, on Nov. 3, 

 at 16** 47™ eq. t, at London,) here, since the mean anomaly is many times 

 greater than -^ of a second, the rule in the second case of the first corollary may 

 be used ; that is, by taking the sine of a = n . 



3t p 



But the number n or^ — m is = 0,05794134 ; and - will be = 0,0030827; 



N 

 P 



therefore (n — - =) 0,0576330/, will be the sine whose arch 3'',30397 is the 

 angle a; and the multiple angle n X a = 10° 26' 53*,05, will be the angle to 

 be first assumed for the anomalia eccentri ; the error of which will be found to 

 be less than a second. 



The true anomaly, computed from this angle according to the rule in the 

 example for Mercury, will appear to be 171° 38' 24", from the perihelion. 



