117 



On Lacaille's Method.* 



The indireft method ufed by Lacaille, and recommended by Lalande, 

 is as follows. 



A fuppofition is made for the true anomaly, with this fuppofed true 

 anomaly the excentric anomaly is computed by the analogy. 

 ^\ — e : j^\+e : : t,\ anom. : t, i excent. anom. 



From the excentric anomaly fo deduced, the mean anomaly is com- 

 puted by the equation m = c-\-es, c. The error of this mean anomaly 

 is applied as a correftion to the affumed anomaly, and the operation 

 repeated till no error remains in the computed mean anomaly. The 

 proper mode of eftimating the value of this method feems to be to 

 enquire how the repeated operations converge. 



To afcertain the rate of convergency, we may make ufe of the 



equation a = fn — 2es, m+ &c. Let »i be the mean anomaly, computed 



to the affumed anomaly a, fuppofing a nearly equal a. 

 Then 



a — fl=?« — m — 2es, ?n — s, m &c. = m-?n x i - iecsin, &c. 



fo that the error of a differs from the error of m 



only by m — mx2ecs, m 



Hence the convergency proceeds according to the fnnple power of the 

 excentricity, and if the mean anomaly be the firll affumption, the error 



of 



* Mem. Acad. 1750. Lalande's Aftr. Lalande edit. Halley's Tab. vol. *, 50. 



