119 



city the error of c will depend only on the feventh &c. Hence in 



orbits of fmall excentricity the excentric anomaly is readily obtained^ 

 And even in orbits of the greateft excentricity, Caffini's firfl approx- 

 imation is fufficient to apply the above formula for a further correc- 



tion. For taking the extreme cafe, viz. when e=l; m-\-m:=c+c-^s, c-\-c 



\ I I 



I \ \ 

 )n=c-\-s, c. From whence is readily deduced 



\ 



»2=C+a,CXC— zC +&C. S, CX'C'' jsC" +&C. 



I I I I \ t 



Now the firfl; approximated excentric anomaly with this excentricity is 

 half the mean anomaly, and its greateft error will be when the mean 



3>'4«S9 ., , . , 



anomaly = 180°. In that cafe c = — =1,57079, with which 



value of c both the feries i — ^^^ + &c. W - i\c''-^kc. will converge 



/ 



;fo that a near value of c may be^derived from the equation ra=c+ca,c 



I ' ' ' t I \ 



m 



or from c= ; But when m=i8o8 the approximated excentric ano- 



' I -\-cs,c 



maly will continually approach to i8o° alfo, and confequently the denomi- 



nator i-j-ac become evanefcent. In this extreme cafe the formula from the 

 fimple equation fails, but a formula might if it were worth while be here 



obtained 



