Olbers's Method of determining the Orbits of Comets, 203 



of Lagrange) were believed to have demonstrated the impossi- 

 bility of solving, generally, and in finite terms, equations of 

 higher degrees than the fourth; an idea which has been am- 

 ply borne out by the results at which I have subsequently ar- 

 rived. I am, Gentlemen, yours, &c, 



West Park, Bristol, June 12, 1835. G. B. JERRARD. 



XXV. On Olbers's Method of determining the Orbits of 

 Comets. By Professor Encke. 

 [Continued from p. 132.] 

 FT only remains now to determine the elements of the orbit 

 * from the known quantities p, p" 9 r, r", k. One might use all 

 these five quantities and obtain convenient expressions for 

 every single element. But here, likewise, the way recom- 

 mended by Gauss, who only employs the first two quan- 

 tities p and p" 9 seems to be more suited to the purpose. 

 In this manner, without making the calculation much longer, 

 the great advantage is gained of having a sure check on the 

 accuracy of all preceding calculations, as the values thus found 

 must exactly accord with the given data. The formulas which 

 Gauss proposes are, again, adapted to the use of his logarithmic 

 tables, and exceedingly convenient in their application. I shall, 

 however, add to some of them other formula?, of which the 

 Theoria Motus presents such a number, which do not require 

 the use of these tables, but seem for that very reason to be in- 

 ferior in respect of the accuracy of the numerical results, on 

 account of the uncertainty of the last figure of the logarithms. 



Gauss designates 

 The heliocentric longitudes of the comet at the first"1 „ 



and third observations by J 



The heliocentric latitudes by /3 /3" 



The longitudes on the orbit by v u" 



The longitude of the ascending node by ft 



The inclination of the orbit, which, according to the'] 



usual distinction of the direct and retrograde motion )> i 



of comets, is always less than 90° J 



The longitude of the perihelion cd 



The time of passage through the perihelion T 



The distance in the perihelion q 



The following calculations must be made in succession : 

 u -f g cos <p 

 P = h~^~ 



p«=Mp. A$ 



2D2 



