194 



MOTION UNDER CONSTANT FOECES 



reaches such a size that it touches the fixed surface at some point 

 P, one of the beads has arrived at this surface and, moreover, has 

 arrived in shorter time than any of the others. Thus it has found 

 the quickest path from to the surface. This path is OP, and 

 we can now fix the path without performing the experiment, from 

 the knowledge that a sphere drawn so as to have as its highest 

 point, and to pass through P, must touch the surface at P. 



In the same way, if we wish to 

 find the quickest time from a surface 

 to a fixed point below it, we have 

 to find a sphere which touches the 

 surface at some point P and has for 

 its lowest point. Then PO will be the 

 path required. For it is seen at once 

 that the time down all the chords of 

 this sphere which pass through is 

 the same, so that the time down PO 

 is equal to the time down any other 

 chord QO, and therefore less than the time down the complete 

 path Q'O from the surface to 0, of which the chord QO is a part. 



ILLUSTRATIVE EXAMPLE 



A ship stands some distance from its pier, 

 and it is required to place a chute at some point 

 of the ship's side, so that the time of sliding 

 down the chute on to the pier may be as short 



as _ 



Clearly the lower end of the chute must just 

 rest on the nearest point O of the pier, and the 

 problem reduces to that of drawing a sphere to 

 have as its lowest point and to touch the ship's 

 side. Assuming the ship's side to be vertical, 

 the tangents to this circle at the ends of the 

 chute must be horizontal and vertical ; whence it 

 is easily seen that the chute must be placed so ) 

 that it makes an angle of 45 with the vertical. 



FIG. 109 



