102 y -TOWTOm s PRINCIPIA. 



sphere) and ; and from that distance S f 



two cycloids touching the sphere, or its great circle, and 



meeting in the point so found. If to that point S, a flexible 

 line or thread be attached and bent round one of the 

 cycloids SP, it will unrol itself and then bind itself 

 round the other cycloid S P , and its extremity will de 

 scribe the cycloidal curve P P required, one of whose 

 properties is, that all the vibrations in its arches are per 

 formed in equal times, however unequal the lengths of 

 these arcs may be, provided that the centripetal force is in 

 each part of the curve directly as the distance from the 

 centre, and that no other force acts on the moving body. 

 But the same solution may be generalised and applied 

 to any given curve whatever ; for the curves found, and 

 along which the flexible line is traced and from which it is 

 then unrolled, are the evolutes of the given curve; and are 

 found in each case by means of the radius of curvature, 

 being the curve formed by its extremity, or the locus of the 

 centres of the osculating circles to all the given curve s 

 points. If the curve in which the body is to move be a 

 circle, the evolute is, of course, a point, the centre of 

 that circle, the radius of curvature being that of the circle. 

 If the curve is a conic parabola, it will be found that the 

 evolutes, or the lines from which the pendulum s thread 

 must wind off, are cubic parabolas, whose equation is 

 ?/ 2 = (!#) 3 , the length of the pendulum being unity. The 

 only case of the problem investigated by Sir Isaac Newton 

 is that of the cycloid, which has the remarkable property. 



