MR CLERK MAXWELL ON THE THEORY OF ROLLING CURVES. 527 



not in the straight line, the calculation is somewhat complex, we shall therefore con- 

 sider only the relations between the curves described in the first and second cases. 



Definition. — The curve which cuts at a given angle all the circles of a given 

 radius whose centres are in a given curve, is called a tractory of the given curve. 



Let a straight line roU on a curve A, and let a point in the straight line de- 

 scribe a curve B, and let another point, whose distance from the first point is b, 

 and from the straight line a, describe a curve C, then it is evident that the curve 

 B cuts the circle whose centre is in C, and whose radius is b, at an angle whose 



sine is equal to -, therefore the curve B is a tractory of the curve C. 



When a = b, the curve B is the orthogonal tractory of the curve C. If tan- 

 gents equal to a be drawn to the curve B, they will be terminated in the curve C ; 

 and if one end of a thread be carried along the curve C, the other end will trace 

 the curve B. 



TNTiea a = 0, the curves B and C are both involutes of the curve A, they are 

 always equidistant from each other, and if a circle, whose radius is b, be rolled on 

 the one, its centre Avill trace the other. 



If the curve A is such that, if the distance between two points measured along 

 the curve is equal to b, the two points are similarly situate, then the curve B is 

 the same with the curve C. Thus, the curve A may be a re-entrant curve, the 

 circumference of which is equal to b. 



When the curve A is a circle, the curves B and C are always the same. 



The equations between the radii of curvature become 



1 1^ r 



T "*" »-2 ~ a Rj 



When a = 0, T = 0, or the centre of curvature of the curve B is at the point 

 of contact. Now, the normal to the curve C passes through this point, therefore — 



" The normal to any curve passes through the centre of curvature of its trac- 

 tory." 



In the next case, one curve, by rolling on another, produces a straight line. 

 Let this straight line be the axis of y, then, since the radius of the rolled curve is 

 perpendicular to it, and terminates in the fixed curve, and since these cuiTes have 

 a common tangent, we have these equations, 



d y, d 6, 



^ ^ d Xj dr^ 



d d.j 



If the equation of the rolled curve be given, find ~ in terms of r , substitute 



VOL. XVI. PART V. 6 U 



