113 The Bight Hon. Lord Kelvin [May 12, 



point moving in a plane under the influence of a given conservative 

 system of forces, that is to say, such a system that when material 

 particles not mutually influencing one another are projected from one 

 and the same point in different directions, but with equal velocities, 

 the subsequent velocity of each is calculable from its position at any 

 instant, and all have equal velocities in travelling through the same 

 place whatever may be their directions. The theorem of curvature, 

 of which I told you in connection with the railway engineering 

 problem, is now simply the well-known elementary law of relation 

 between curvature and centrifugal force of the motion of a particle. 



The motion of a particle in a plane is, as Liouville 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 problem of dynamics involving only two freedoms to 

 move, I am sure you will understand sufficiently to admire the great 

 beauty of this theorem. 



The doctrine of isoperimetrical problems in its relation to dy- 

 namics is very valuable in helping to theoretical investigation of an 

 exceedingly important subject for astronomy and physics — the stability 

 of motion, regarding which, however, I can only this evening venture 

 to show you some experimental illustrations. 



The lecture was concluded with experiments illustrating — 



1. Rigid bodies (teetotums, boys' tops, ovals, oblates, &c.) placed 

 on a horizontal plane, and caused to spin round on a vertical axis, 

 and found to be thus rendered stable or unstable according as the 

 equilibrium without spinning is unstable or stable. 



2. The stability or instability of a simple pendulum whose 

 point of support is caused to vibrate up and down in a vertical line, 

 investigated mathematically by Lord Ray lei gh. 



3. The crispations of a liquid supported on a vibrating plate, 

 investigated experimentally by Faraday ; and the instability of a 

 liquid in a glass jar, vibrating up and down in a vertical line, demon- 

 strated mathematically by Lord Rayleigb. 



4. The instability of water in a prolate hollow V6S661, and its 



