Prof. Sylvester's Astronomical Prolusions. 55 



continue to obtain where, e being greater than 1, the motion 

 becomes hyperbolic. If fju be the absolute force, and 1, as before, 

 the semi-major axis, writing 



V e+. 



— p etang=#, etan^ =a/, 



the rest of the notation being preserved, we obtain, by direct 

 integration and substitution, 



s_ 1 / e 2 + a? g e 2 4-a/ 2 \ 

 2~l-e 2 Vl-^ + T^F/' 



A_ _1 (e« + g«)(e» + s' g ) 



8 " (1-e 2 ) 2 (l-ar s )(l~*«) ' 



And we must needs find by actual computation 



the Jacobian jf& A ' 5 = . 

 d[€j oc, a?) 



Making e=0, and giving #, a/ their corresponding values in 

 terms of s and A, there results 



a? 2 s + c a/ 2 





1-^2 2 ' l-#' 2 2 

 and accordingly 



^ flog(\A-|-c + 2 — \A— c) — log(\A— c— 2— \/s— c) 

 "^l -v/(« + c+l) 2 -l + v/(s-c+l) 2 -l. 



It is worthy of notice that the effect of making e=0 or 

 e= — 1 in this case, like that of making e=l in the case of 

 elliptic motion, is to reduce the motion to that of a body in a 

 straight line, but with the difference that for the elliptic the 

 reduced motion is that of a body moving between the point of 

 instantaneous rest and the centre of force or point of infinite 

 velocity, whereas for the hyperbola it is that of a body moving 

 on the same side of these two points. 



The theorem for the case of the parabola was given by Euler 

 (1744), but reproduced independently by Lambert in the Insig- 

 niores Proprietaries, Sectiones 1, 2, in 1761. 



I think the idea of the general theorem may not unlikely have 

 originated in an observation of the accordance in form of the 

 result for parabolic motion with that for motion in a straight 

 line, an accordance easily verified to extend to motion in a circle. 



