520 MB CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 



In the volume for 1 707, page 79, there is a paper entitled,—" Methode gene- 

 rale pour determiner la nature des Courbes formees 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, — 



1st, When the tracing-point is the origin of the rolled curve. 

 2d, When the fixed curve is the same as the rolling curve. 

 3<:/, When both of these conditions are satisfied. 

 4:th, When the fixed Hue 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 ex- 

 amples 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 Princijdes of Merhanism. Class C. Rolling Contact. 



He employs the same method of finding the one curve from the other which 

 is used here, and he attributes it to Euler (see the Acta Petvopolitava, vol. v.). 



Thus, nearly all the simple cases have been treated of by different authors ; 

 but the subject is still far from being exhausted, for the equations have been ap- 

 plied to very few curves, and we may easily obtain new and elegant properties 

 from any curve we please. 



Almost all the more notable curves may be thus linlied together in a great 

 variety of ways, so that there are scarcely two curves, however dissimilai-, be- 

 tween which we cannot form a chain of connected curves. 



This will appear in the list of examples given at the end of this paper. 



Let there be a curve KAS, whose pole is at C. 



Let the angle DCA=6, and CA=r, and let 



Let this curve remain fixed to the paper. 



Let there be another curve BAT, whose pole is B. 



Let the angle MBA=&.., and BA=r,,, and let 



