4 MEADOWCROFT, Motion in Elliptic and Hyperbolic Orbits 



Similar considerations may be applied to obtain corresponding 

 results for the hyperbola, but the results are necessarily more complicated 

 as the representation of points on the curve by means of a single para- 

 meter is not of so simple a character as in the ellipse. If the equation 



of a hyperbola is — - — = i the co-ordinates of any point can be 

 a 1 b 2 



represented by a single parameter,^, by the relations x = a cosh ^, 



y = b sinh p. If, in the figure, P is the point (x, y), corresponding to 



the parameter ^ then CN= a cosh /x , PN— b sinh ^, and it is easily 



shown that the area of the sector CPA is equal to \ abu. 



Adams'* theorem for the hyperbola may be enunciated as follows : — 

 " If / is the time of describing any arc P T P 2 of a hyperbola, and k is the 



chord of the arc, then / j&- = - ^ -j- <£ 2 + sinh ( f )I - sinh (f> 2 , 



where sinh \ <£, = \ I— — — — , sinh \ ^ 2 = \ ,— — 



■k" 



J' 



'British Association Report, 1877, or Collected Works, p. 410. 



