Prof. Sylvester's Astronomical Prolusions. 



54 



Denoting this determinant by 



A , B , C 



D ; E , F 

 G , H , K 



we find 



so that J = 



(A, B, C)-2H(D, E, F) + 2E(G, H, K) = (0, B,-B), 

 (A, B, C) -2K(D, E, F) f 2F(G, H, K) = (0,-C, C), 



A, B, C 



=0. 



0, B,-B 

 0,-C, C 



Hence it appears that da> is a linear function of ds and c?A; 

 that is, w is a function of s and A, or, what is the same thing, 

 of s and c, and independent of e. If then, when e=l, the cor- 

 responding values of p, p 1 , v, v\ u, v! are r, r 1 , 0, 6', <p, </>', we 

 have cos0= — 1, cos0'=— 1, sin 0=0, sin0'=O, r—r 1 =c } 

 r+r'=s, whence, writing 



writing 



1 — COSp = — gr> 



1 J.I s ~~ c 



we have finally &)=<£— <f> ! — sin <£+ sin <j>', as was to be proved. 



Essentially this demonstration is of the same nature as the 

 first of Lagrange's four methods of proof above referred to, 

 but with the difference that it leads up to and accounts 

 beforehand for the success of the transformations therein em- 

 ployed. 



Alluding to Lambert's cumbrous demonstration, Lagrange says 

 of it, "His theorem merits the especial notice of mathematicians, 

 both on its own account, and because it appears difficult to arrive 

 at it by algebraical processes (calcul) ; so that it may be ranked 

 among the small number of those in which geometrical seems 

 to have the advantage over algebraical analysis." In the nature 

 of things such advantage can never be otherwise than temporary. 

 Geometry may sometimes appear to take the lead of analysis, 

 but in fact precedes it only as a servant goes before his master 

 to clear the path and light him on his way. The interval between 

 the two is as wide as between empiricism and science, as between 

 the understanding and the reason, or as between the finite and 

 the infinite. 



The result so simply obtained above is of course not restricted 

 to the case of the ellipse, but applies to motion generally about a 

 centre of force according to the law of nature. 



Calling t the time, the syzygy shown to exist between o7, 8s, 

 Be, being independent of any supposition as to the value of e, 

 or as to the reality of the functions employed, will of necessity 



