250 CENTRAL FORCES; 



If we now compare the motion of different bodies in con- 

 centric orbits of the same conic sections, we shall find that 

 the areas which, in a given time, their radii vectores describe 

 round the same focus, are to one another in the subduplicate 

 ratio of the parameters of those curves. From this it follows, 

 that in the ellipse whose conjugate axis is a mean propor- 

 tional between its transverse axis and parameter, the whole 

 time taken to revolve (or the periodic time / being in the pro- 

 portion of the area (that is in the proportion of the rectangle 

 of the axes) directly, and in the subduplicate ratio of the 

 parameter inversely, is in the sesquiplicate ratio of the trans- 

 verse axis, and equal to the periodic time in a circle whose 

 diameter is that axis. It is also easy to show from the 

 formula already given respecting the perpendicular to the 

 tangent, that the velocities of bodies moving in similar conic 

 sections round the same focus, are in the compound ratio of 

 the perpendiculars inversely and the square roots of the 

 parameters * directly. Hence in the parabola a very simple 

 expression obtains for the velocity. For the square of the 

 perpendicular being as the distance from the focus by the 

 nature of the curve (the former being a 2 + a x, and the latter 

 a + .r), the velocity is inversely as the square root of that 

 distance. In the ellipse and hyperbola where the square of 

 the perpendicular varies differently in proportion to the dis- 

 tance, the law of the velocity varies differently also. The 

 square of the perpendicular in the ellipse (A being the trans- 

 verse axis and B the conjugate, and r the radius vector) is 



B 2 X r B 2 X r 



; in the hyperbola, , or those squares of the 



T T 



perpendicular vary as and , in those curves re- 



A - r A + r 



spectively, B 2 being constant. Hence the velocities of bodies 

 moving in the former curve vary in a greater ratio than that 



* By parameter is always to be understood, unless otherwise mentioned, 

 the principal parameter, or the parameter to the principal diameter. 



