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XXXV.—On the Theory of Rolling Curves. By Mr James CLerK MAXWELL. 
Communicated by the Rev. Professor KELLAND. 
(Read, 19th February 1849.) 
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 atten- 
tion 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 meansanew one. The first idea of the 
_ eycloid is attributed to ARISTOTLE, and involutes and evolutes have been long known. 
In the “ History of the Royal Academy of Sciences” for 1704, page 97, there 
is amemoir entitled “ Nouvelle formation des Spirales,” by M. Varianon, in which 
he shews how to construct a polar curve from a curve referred to rectangular co- 
ordinates by substituting the radius vector for the abscissa, and a circular arc for 
the ordinate. After each curve, he gives the curve into which it is “ unrolled,” 
by which he means the curve which the spiral must be rolled upon in order that 
its pole may trace a straight line; but as this is not the principal subject of his 
paper, he does not discuss it very fully. 
There is also a memoir by M. pE La Hire, in the volume for 1706, Part I., 
page 489, entitled,—“ Methode generale pour réduire toutes les Lignes courbes a 
des Roulettes, leur generatrice ou leur base étant donnée telle qu’on voudra.” 
M. pE LA Hire treats curves as if they were polygons, and gives geometrical 
constructions for finding the fixed curve or the rolling curve, the other two being 
given ; but he does not work any examples. 
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WiOli XV. PART, Vi, 
