"3 



I, 2 



c is never greater than — -— and the errors after each operation will be 



nearly proportional toe , c , c , &c. But the limit very nearly of the 



error of ^ after the firft operation may be eafily obtained. For the error of 

 \ 



c is lefs than s xt ,a therefore the limit of the error of 



280 



c= x-f X ^ 5 ax nearly.This quantity is a maximum when 



^ 56X280 I — cs, c 



c= 1 69° nearly, in which cafe the quantity itfelf =,00001 52=3* nearly. 



Aclual computation fliews the error in this cafe to be 1" nearly. 



Hence the following obfervations on Machin's folution of Kepler's 

 problem. 



1. His method of obtaining a firft approximation is general for all 

 elliptic orbits whatever, and will give the excentric anomaly always fuf- 

 ficiently near for correfting it by his fecond rule. The greateft error 

 is i". i3'5-, viz. when the ellipfe is evanefcent, and the mean anomaly 

 = 180°. When the excentric anomaly is lefs than 23° the ufe of the 

 fecond rule is unneceffary, for then the firft approximation gives the ,ex- 

 centric anomaly to lefs than a fecond. 



2. It will be very rarely neccfiary to repeat the fecond rule, for the 

 excentric anomaly found by one operation will never err more than o." 

 from the truth, and that only in the extreme cafe, viz. when the el- 

 lipfe is evanefcent and the excentric anomaly 169". 



3. This folution, therefore, of Kepler's problem is complete, but the 

 praftice of it, particularly as given by the author, is not fo convenient 

 in the planetary orbs as other methods. 



4. In the extreme cafe, viz. when- e= i and therefore the ellipfe is eva- 

 nefcent, this mode of folution becomes fimpler than in any other cafe, for 



Vol. IX. ( P ) ' then 



