136 AdwuoniicnJ and Ntiiitiral Collections. 



3 e/' —P) - ^\ fe' + 3 e c'— /'J since e' —/' - c' ; but 

 e^ + 3ec' = iCe + cf + iCe — cf; and since /= = e' — c' 

 = re + cj (e — c), e^ + 3 e 0=—/" = | r[e + c^ + fe — c;'— 2 



fe + c)" (e — cj -, and the square of the area divided by 



2? is TjL ( [e + c] - — (e — cj'^ J , consequently the area itself 



Cor. The time of describing a circle at the mean distance of the 

 earth being called unity, and the time occupied by a comet of 

 the same perihelion distance being less in the ratio of 1 to 

 -v/ 2, the time for such a comet will be expressed by the area 



described, divided by 3.1416 ^/ 2, that is, by-j^^ ({e-\- cY — 



[e — c]^) ; and since the area described is in the subduplicate 

 ratio of the parameters, the same formula will express the 

 time in all other cases ; the units being the year and the earth's 

 mean distance from the sun.] 



§9. 

 If we consider these four equations with a little attention, we 

 shall soon be convinced that it is perfectly impossible, in the 

 present state of analytical science, to determine the three un- 

 known quantities, §', f, and §'", immediately from them. For, 

 supposing the patience of the calculator to be even sufficient to 

 develope the equations completely, to free them from surds, and 

 to substitute for r, k, x, y, and z, their values in terms of §, he 

 woxdd still arrive at equations of so high a degree, in which the 

 three unknown magnitudes, or at least two of them, if one were 

 exterminated by means of the first equation, are involved with 

 each other, that he could obtain no results whatever from them; 

 and on this involution depends the insuperable difficulty of the 

 problem. If, indeed, the second equation were as simple as the 

 first, and if it enabled us to exterminate another of the unknown 

 quantities, so that one only remained, we should easily find 

 means to resolve the last two equations'in a convenient manner, 

 were they even still more intricate than the results of the elegant 

 theorem of Lambert; and in this case it would be possible to 



