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



We have the equations x = px, 



y = -w> 



and we deduce that x and y must be given by equations of the 



form 



x = A cos (t V/t*) + B sin 



= G cos "- + D sm 



where A, B, C, D are arbitrary constants depending on the initial 

 conditions, viz. A and C are the coordinates, and B>Jfi, 

 the resolved velocities at the instant t = 0. 



Solving the above equations for cos (t /vV) and sin (t ^/JL), we have 

 (AD - BC) cos (* V/*) = -0* - % (4D - 0) sin 

 eliminating , we find 



so that the path of the moving point is an ellipse whose centre 

 is the origin, and whose position with reference to the origin and 

 axes is fixed. The whole motion is clearly periodic with period 



** 

 Jp* 



Let us change the axes to the principal axes of the ellipse, 

 and suppose the moving point is at one extremity (x = a) of the 

 major axis at the instant t = 0, then at this instant x = a, y = 0, 

 and, since the point is moving at right angles to the major axis, 

 x = 0. Suppose y = 6 *J/JL. Then we must have at time t 



a; = a cos (t V^), y = b sin (t VA*) 



Thus 6 is the semi-axis minor, and t *Jp is the eccentric angle at 

 time t. 



The point therefore moves so that its eccentric angle in- 

 creases uniformly with angular velocity 



49. Examples. 



1. Prove that when the equation is x=:pj}, where p is positive, and the 

 initial conditions are that X=X Q and x=x when t=0, then at any time t 



X=X Q cosh (t V/i) + ~,~ sinn (t \W 

 VM 



2. Prove that when the acceleration is directed from the origin and is 

 proportional to the distance the path is an hyperbola. 



