[From the Transactions of the Royal Society of Edinburgh, Vol. xvi. Pan \ .] 



II. On the Tlicory of Rolling Curves. Communicated by the Rev. Professor 



KELLAND. 



THERE is an important geometrical problem which proposes to find a curve 

 having a given relation to a series of curves described according to a given 

 law. This is the problem of Trajectories in its general form. 



The series of curves is obtained from the general equation to a curve by 

 the variation of its parameters. In the general case, this variation may change 

 the form of the curve, but, in the case which we are about to consider, the 

 curve is changed only in position. 



This change of position takes place partly by rotation, and partly by trans- 

 ference through space. The rolling of one curve on another is an example of 

 this compound motion. 



As examples of the way in which the new curve may be related to the 

 series of curves, we may take the following : 



1. The new curve may cut the series of curves at a given angle. When 

 this angle becomes zero, the curve is the envelope of the series of curves. 



2. It may pass through corresponding points in the series of curves. 

 There are many other relations which may be imagined, but we shall confine 

 our attention to this, partly because it affords the means of tracing various 

 curves, and partly on account of the connection which it has with many 

 geometrical problems. 



Therefore the subject of this paper will be the consideration of the relations 

 of three curves, one of which is fixed, while the second rolls upon it and 

 traces the third. The subject of rolling curves is by no means a new one. 

 The first idea of the cycloid is attributed to Aristotle, and involutes and 

 evolutes have been long known. 



