I20 



obtained from the quadratic equation m=fx I -fac-^ c s,c. In this man- 



ner, therefore, a complete folution of Kepler's problem may be obtain- 

 ed, and very conveniently, except when the cllipfc is evanefcent, and 

 at the fame time the mean anomaly nearly i8o°*. 



When the ellipfe is very excentric or nearly evanefcent, and the 

 mean anomaly nearly i8o°, a fmall error in the excentric anomaly oc- 

 cafions a great error in the true anomaly. Hence an inconveniency 

 in this cafe, in deriving from the excentric anomaly, computed by any 

 method, the true anomaly. This takes place with refpeft to comets, 

 when they are near perihelion or are vifible to us. And therefore, for 

 them, when near perihelion, inftead of ufing the following praftical 

 rule, the beft method is to derive by a correftion the true anomaly, 

 reckoned from perihelion from the anomaly in a parabola, having the 

 fame perihelion diftance. 



I 

 The Log. of the multiplier , =: 20 — log. 2 — 2 log. cs\a., 



1+ ecs^c 



K. being an arch the log. cofine of which is the Log. e -{- Log. a, c. 

 It appears to me rather more convenient in practice to compute pre- 

 vioufly a fmall table containing the logarithms of the above multipliers, 

 which may be done very expcditioufly for a given orbit. In the moll 

 excentric planetary orbits, if the table be computed to every five de- 

 grees of excentric anomaly, and the logarithms to five places of 

 figures, it will be fufficient. That a comparifon may be made of the 

 praftice of the method with, and without the tables, the table is 

 fubjoined for the new planet Pallas difcovered by Dr. Olbers. 



What has been faid will, I think, fuiEciently explain the convenience 



and extent of the following praftical rule. 



PraSlkal 



• This reafoniog may be eaCly applied to Newton's firft folution, and in the ex- 

 amination of which it might have been inferted, but it was thought better to place it 

 here, becaufe Caffini's folutioo was pofterior to Newton's. 



