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XXXV.— Ow 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 cm-ves 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 Avay in which the new curve may be related to the series 

 of cm-ves, we may take the following : — 



1. The new cm-ve may cut the sei-ies of curves at a given angle. When 

 this angle becomes zero, the curve is the envelope of the series of cm'ves. 



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 

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

 third. The subject ofrolling 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. 



In the " History of the Royal Academy of Sciences" for 1704, page 97, there 

 is a memoir entitled " Nouvelle formation des Spu-ales," byM. Varignon, 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. de la Hire, in the volume for 1706, Part II., 

 page 489, entitled,—" Methode generale pour reduire toutes les Lignes courbes a 

 des Roulettes, leur generatrice ou leur base etant donnee telle qu'on voudra." 



M. DE 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. 



VOL. XVI. PART V. 6 S 



