MOTION ON AN INCLINED PLANE 



193 



wires make all possible angles with the vertical, one of them, 00', 



being vertical. Let us imagine that the beads are all collected at 



and are set free simultaneously. 



After time t, let the bead which is 



falling vertically be at P, and let the 



bead which is falling along a wire 



inclined at an angle {3 to the vertical 



be at Q. This latter bead moves with 



acceleration g cos 0. Thus OP = \ gf, 



while OQ = \g cos /3 f. Hence 



OQ=OPcos{}, and therefore OQP 



is a right angle. It follows that Q 



is on the sphere constructed on OP 



as diameter, and obviously the same 



will be true of every other bead. 



Thus at any instant all the beads 



will be on a sphere of which is the highest point, and of which 



the lowest point is at a distance \gt* below 0. Hence as the 



motion proceeds the beads will appear to form a sphere which 



continually swells out in size, the highest point appearing to 



remain fixed at 0, while the lowest point appears to fall freely 



under gravity. 



160. This imaginary experiment 

 indicates a way of solving a prac- 

 tical problem. Suppose we wish 

 to place a smooth plane or wire in 

 such a position that a particle will 

 pass down it from a fixed point 

 to a given fixed surface in the least 

 time possible. Let us suppose that 

 we fix the apparatus of wires and 

 beads at 0, that we set the beads 

 free simultaneously, and watch the 

 increase in size of the sphere which 

 FIG. 107 they form. As soon as the sphere 



