234 ^ ^^''^' ""'^ UNIVERSJL SOLUTION 



It is to be kept in mind, thnt the arch found by this rule is 

 only the firfl: term of a feries of approximations, converging 

 very fail; to the eccentric anomaly : and that, by a repetition of 

 the calculation, a rcfnlt may be obtained, that will fatisfy the 

 moft fcrupulous accuracy. But the rule may be confidered as 

 exadl, as to any real pradlice, for the orbits of all the planets, 

 excepting Mercury : and, even in the orbit of Mercviry, the er- 

 ror will never exceed a few feconds. Let it be obferved, fur- 

 ther, that the error of the rule is chiefly in the arch -I : for the 

 error of i- is of the fame order with the error of e ; whereas the 

 error of <p is of the fame order with the error of e''. 



Example. Let it be required to find the eccentric anomaly, 

 correfponding to the mean anomaly 64° 37' 8".5 in the orbit of 

 Mars, fuppoflng the eccentricity to be rr .093088. 



We have here m — 64° 37' 8".^, and i =: .093088. 

 I. To compute r from the formula 2r — s fin /// ; 



log. £ — 2.9688937 

 log. fin in = 9-9559089 

 conftlog. = 3^5362739 



log. 2r in minutes =: 2.4610765 ; 2r = 289'; andr = 2° 24'. 



2. Then e — s ; and fin 9' = fin m + e- fin m cof ' w. 



Sin T = 8.6219616 



log. £ = 2.9688937 



confl. log. = 3.53'^2739 



fum — 10 r= 1.1271292 



fubtrad log. 144' = 2.1583625 



log. e — 2.9687667 



log. r = 3-9375334 



log. fin m zz 9.9559089 



2 log. cof /« zz 19.2642510 



log. f' fmw/ cof ' m zz. 3'iS7^933i ^^^ ^^ ^^^ '■" cof °«? = .0014378. 



To 



