56 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



Let &amp;lt;, = L QCA in the figure, be the eccentric angle of P, and $, = L ASP, 

 the vectorial angle. 



Then curvilinear area ASP= curvilinear area ANP - triangle SPN 



= - (curvilinear area ANQ}- triangle SPN. 



QJ 



Now curvilinear area AJVQ= sector ACQ - triangle CQN 



= | ( 2 &amp;lt; - a 2 sin $ cos $), 



and triangle SPN= $ b sin &amp;lt;t&amp;gt;(ae-a cos &amp;lt;). 



Hence curvilinear area ASP = db &amp;lt; e sin &amp;lt;. 



Let t be the time from A to P then, since & is twice the area described per 

 unit of time, 



lit = ab (&amp;lt; e sin $). 



a? 



Thus = 4~ (&amp;lt;/&amp;gt; - e sin 0). 



\V 



The quantity vV/ a i g known as the &quot;mean motion&quot; and is denoted by /i, 

 so that the time in question is given by 



Prove that 6 is connected with by the equation 



, ,, , .,. . 



cos&amp;lt;z&amp;gt; = - , and that, if e is small, 

 1 + e cos 6 



0=nt + 2e sin nt approximately. 



7. Two points describe the same ellipse in the same periodic time, starting 

 together from one end of the major axis ; one of them has an acceleration to 

 a focus S, and the other an acceleration to the centre C. Prove that, if (f&amp;gt; 1 and 

 2 are their eccentric angles at any instant, then fa $ 2 =esin( 1 . 



8. Two points describe ellipses of latera recta I and I in different planes 

 about a common focus, and the accelerations to the focus are equal when the 

 distances are equal. Show that, when the relative velocity of the points is 

 along the line joining them, the tangents to the ellipses at the positions of the 

 points meet the line of intersection of the planes in the same point, and that 

 the focal distances, r and /, make with this line angles B and & such that 



r sin 6 r sin & 



55. Motion with a central acceleration varying in 

 versely as the square of the distance. We give here a 

 version of Newton s investigation* of the orbit described by a point 

 which moves from a given position P, with a given velocity F, in 

 a given direction PT, and has an acceleration to a point S varying 

 inversely as the square of the distance from 8. 



* Principia, Lib. I. Sect. in. Prop. 17. 



