

ISOPERIMETRICAL PROBLEMS. 589 



hour longer, I fear that two hours of pure mathe- 

 matics and dynamics might be too fatiguing. I 

 must, therefore, perforce limit myself to the two- 

 dimensional, but otherwise wholly comprehensive, 

 problems of Dido and Horatius Codes. Going 

 back to the simpler included case of the railway of 

 minimum cost between two towns, the dynamical 

 analogue is this : For price per unit length of the 

 line substitute the velocity of a 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 veloci- 

 ties, 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. 



