618] 
535 
618. 
ON THE MECHANICAL DESCRIPTION OF A CARTESIAN. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xm. (1875), 
pp. 328-330.] 
Suppose that in two different curves the radius vectors r, r, which belong to 
the same angle 6, are connected by the equation 
r~ + (Mr + N + ^jr + B = 0; 
then, taking one of the curves to be the circle 
Mr' + = A cos 6, 
r 
the other curve is 
r- + (A cos 0 + N) r + B = 0, 
viz. this is a Cartesian. It perhaps would not be difficult to contrive a mechanical 
arrangement to connect the radius vectors in accordance with the foregoing equation; 
but the required result may be obtained equally well by means of a particular case 
of the relation in question; viz. taking this to be 
r 2 + (— r + N) r + B = 0, 
then, taking the one curve to be the circle / = — A cos 0, ’ the other curve is the 
Cartesian, 
r 2 + (21 cos 0 + B)r + D= 0, that is, r 2 4- (A cos 0 + N) r+B = 0. 
The relation between the radius vectors may in this case be written 
, Ar B 
r= iV + r H—, 
r 
which can be constructed mechanically by a simple addition to the Peaucellier-cell, 
viz. if we joint on to G (fig. 1, p. 536) a rod CD A, having a slot, working on a pin 
at A, so that the rod is thereby kept always in the line BAG, then, making B the