109 



On Machines Method.* 



The accuracy and convenience of the methods hitherto examined ex- 

 cepting the firft method of Newton, depend on the fmall excentricity 

 of the orbit ; and therefore they are only applicable to the planets. 

 They would be ufelefs for the excentric ellipfes in which the comets 

 move. But the problem is otherwife folved for the parts of the or- 

 bits in which the comets are vifible to us. That a complete folution 

 to Kepler's problem might be given, equally applicable to all orbits, 

 Machin propofed his very ingenious method, which does not at all de- 

 pend on the excentricity. For the planetary orbs indeed the method 

 as given by the author, is not fo convenient for praftice as other me- 

 thods. However his method may be rendered confiderably more fim- 

 ple in praftice, as is hereafter pointed at. In examining this method, 

 the method itfelf is firft briefly and fomewhat more fimply ftated, than 

 is done by the author in the Phil. Trans, a limit of the error of the 

 flrfl: approximation is then ftiewn, and alfo the rate of convergency of 

 the fecond approximation. 



The firft approximation may be explained as follows. 



Let m = the mean anomaly (rad. i), c = the excentric anomaly rec- 

 koned from perihelion, for a reafon to be hereafter affigned, the femi- 

 axis major = i and the excentricity = e. 



Then wz = r — es^c, let c =:.na, a being an arch, the fine of which 

 is s and n a number to be determined hereafter. 



Then m — na — es, na. 



Therefore by the feries for the expreifing the arch in terms of the 

 fine, and by the feries for expreffing the fine of the multiple arch. 



* Phil. Trans, vol. 4c. Abridg. vol. 8. 



