590 POPULAR LECTURES AND ADDRESSES. 



The motion of a particle in a plane is, as Liou- 

 ville has proved, a case to which every possible 

 problem of dynamics involving just two freedoms 

 to move can be reduced. But to bring you to see 

 clearly its relation to isoperimetrics, I must tell 

 you of another admirable theorem of Liouville's, 

 reducing to a still simpler case the most general 

 dynamics of two-freedoms motion. Though not 

 all mathematical experts, I am sure you can all 

 perfectly understand the simplicity of the problem 

 of drawing the shortest line on any given convex 

 surface, such as the surface of this block of wood 

 (shaped to illustrate Newton's dynamical theory 

 of the elliptic motion of a planet round the sun) 

 which you see on the table before you. I solve 

 the problem practically by stretching a thin cord 

 between the two points, and pressing it a little this 

 way or that way with my fingers till I see and feel 

 that it lies along the shortest distance between 

 them. And now, when I tell you that Liouville 

 has reduced to this splendidly simple problem of 

 drawing a shortest line (geodetic line it is called) 

 on any given curved surface every conceivable 



