THE THEORY OF BOLLING CURVES. 13 



of the circle ; roll this curve on a straight line, and the centre of the circle will 

 describe a parabola ; roll this parabola on a straight line, and its focus will trace 

 the catenary required. 



We come now to the case in which a straight line rolls on a curve. 



When the tracing-point is in the straight line, the problem becomes that 

 of involutes and evolutes, which we need not enter upon ; and when the tracing- 

 point is not in the straight line, the calculation is somewhat complex ; we shall 

 therefore consider 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 roll on a curve A, and let a point in the straight 

 line describe 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 T, 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 

 tangents 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. 



When 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 

 6, be rolled on the one, its centre will 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 



11 r 



