THE THEORY OF ROLLING CURVES. 5 



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

 there is a memoir entitled "Nouvelle formation des Spirales," by M. 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 n., 

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

 des Koulettes, leur generatrice ou leur base etant donne"e telle qu'on voudra." 



M. de la Hire treats curves as if they were polygons, and gives geome- 

 trical constructions for finding the fixed curve or the rolling curve, the other 

 two being given ; but he does not work any examples. / 



In the volume for 1707, page 79, there is a paper entitled, "Methode 

 generale pour determiner la nature des Courbes formers par le roulement de 

 toutes sortes de Courbes sur une autre Courbe quelconque." Par M. Nicole. 



M. Nicole takes the equations of the three curves referred to rectangular 

 co-ordinates, and finds three general equations to connect them. He takes the 

 tracing-point either at the origin of the co-ordinates of the rolled curve or not. 

 He then shews how these equations may be simplified in several particular 

 cases. These cases are 



(1) When the tracing-point is the origin of the rolled curve. 



(2) When the fixed curve is the same as the rolling curve. 



(3) When both of these conditions are satisfied. 



(4) When the fixed line is straight. 



He then says, that if we roll a geometric curve on itself, we obtain a new 

 geometric curve, and that we may thus obtain an infinite number of geometric 

 curves. 



The examples which he gives of the application of his method are all taken 

 from the cycloid and epicycloid, except one which relates to a parabola, rolling 

 on itself, and tracing a cissoid with its vertex. The reason of so small a 

 number of examples being worked may be, that it is not easy to eliminate 

 the co-ordinates of the fixed and rolling curves from his equations. 



The case in which one curve rolling on another produces a circle is treated 

 of in Willis's Principles of Mechanism. Class C. Rolling Contact. 



