228 J NEW and UNIFERSJL SOLUTION 



The cubic equation therefore becomes x'^ -}- a x -zz b : and it 



is manifefl that x is nearly — fin 72 ° 24 ; and, having cor- 



redled this vahie, by the ordinary method, I find, 



A? — fin I? =: .9534420 



therefore <p 1= 72° 15' 3T". 



„ ^(D -{- m „ (p — m ^ f. , 



But, cof ^- : cof i^ : : cof ip : cof t, 



log. cof <p -=. 9.4839026 

 log.cof^— --= 9.9975102 



19.4814128 

 log. cof^-^t£' = 9.6071052 



log. cof f =: 9.8743076. ., 



Therefore 4^ rz 41° 31' 20", and 



confequently />' = (p — 4" = 30° 44 n'j wlxlch is greater than 

 BH. -"•'■ 



M. EuLER finds the arch AH = 129° 16' 27"; therefore, 

 BH •=. 30° 43' 33" : fo that the fecond approximation p\ differs 

 little more than half a minute of a degree from the truth. 



14. The only cafes of Kepler's problem that are interefling 

 to the aftronomical obferver, are the two extreme cafes, when 

 the eccentricity is very fmall, and when it is very great, ap- 

 proaching to unity. The former of thefe two cafes is that of 

 the planets, all of which defcribe orbits very little eccentric, and 

 nearly circular ; the latter is that of the comets, which, on the 

 contrary, move in very eccentric orbits. The principal obje<5l 

 of this paper is accompliihed in what has already been done ; 

 but it yvill be no improper fequel, to apply the general method 

 to the two cafes juft mentioned, -j '^':'' '' '"'' 



The fuppofition that the eccentricity is fmall, contributes 

 greatly to remove the chief difficulties that occur in the folution 



of 



