234 A NEW and UNIVERSAL SOLUTION 



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

 only the firfb term of a feries of approximations, converging 

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

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

 moft fcrupulous accuracy. But the rule may be confidered as 

 exacl, as to any real practice, for the orbits of all the planets, 

 excepting Mercury : and, even in the orbit of Mercury, 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 ^ : for the 

 error of 4* is of the fame order w T ith the error of e ; whereas the 

 error of P is of the fame order with the error of e 1 . 



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

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

 Mars, fuppoiing the eccentricity to be zz .093088. 



We have here m zr 64 37' 8".5, and g zz .093088. 

 1. To compute r from the formula 2r — s fin m ; 



log. s — . 2.9688937 

 log. fin m — 9.9559089 

 conftlog. = 3-5362739 



log. 2r in minutes zz 2.4610765; 2t — 289'; andr = 2° 24'. 



r 



2. Then e zz. s ; and fin 9' zz fin m + <? 2 fin m cof 2 m 



r 



Sin r == 8. 62 1 96 1 6 



log. g — 2.9688937 



conft. log. = 3»53 62 739 



fum — ioz 1.1271292 



fubtratfl log. 144' == 2.1583625 



log. e = 2.9687667 



log. e* - 3-9375334 



log. fin m zz 9.9559089 



2 log. cof m zz 19.2642510 



log. t x fin;/z cof' m zz 3*1576933, and e 2 fin m cof 2 m zz .0014378. 



To 



