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,
pQR' = dQR.
Fig. 31.
The three other groups of equations are similarly obtained
rQ’S' = aQS, qP'S' = cPS, rP'S' = bPS,
qR'S' = aRS, pQ’S1 = cQS, pR'S' = bRS,
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, anti that consequently for any two
points on the circle P and Q, with their corresponding points P’ and Q', we
must have
P'Q'
-PQ‘