150] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 139 



velocities must be proportional to the sides of the triangle P Q R . 

 Introducing another quantity d, we have 



rP Q = dPQ, 

 qP R = dPR, 



The three other groups of equations are similarly obtained 

 rQ S = aQS, qP S = cPS, rP S = bPS, 



sR Q = aRQ, sQ P = cQP, sR P = bRP. 



Whence we easily deduce 



ap = bq = cr = ds = hpqrs, 



where h is a new quantity. We hence obtain from the first equation 



P Q = hPQpq. 



As this is absolutely independent of R and S, it follows that h must be inde 

 pendent of the special points chosen, and that consequently for any two 

 points on the circle P and Q, with their corresponding points P and Q , we 

 must have 



P Q 



PI 



