35] KINETIC FOCI. 105 



intersection of courses whose angles of elevation a differ infinitely 

 little. If the second point x lt y^ lie without this envelope it cannot 

 be reached under the given conditions. If upon it it can be reached 

 by one path, and if within it by two paths. In that case the course 

 that reaches x lf y before touching the envelope has the less action. 

 A point at which two infinitely near courses from a given point 

 with equal energy intersect is called a kinetic focus of the starting 

 point, and if on any course the terminal configuration is reached 

 before the kinetic focus on that course, the action will be a minimum. 

 If the kinetic focus is first reached it will not. 



Thus in the problem of motion on a sphere under no forces, 

 the point diametrically opposite the initial point is a kinetic focus. 

 Evidently a particle may reach the kinetic focus starting in any 

 direction from the original point, for all great circles through a 

 point intersect in its opposite point. The envelope of all the great 

 circles or courses from a point in this cases reduces to a point, which 

 is the kinetic focus. 



For the treatment of the difficult subject of kinetic foci, which 

 belongs to the calculus of variations, the reader is referred to 

 Thomson and Tait, Principles of Natural Philosophy, 358, and 

 Poincare, Les Methodes Nouvelles de la Mecanique Celeste, Tome III, 

 p. 261, also to Kneser^ Lehrbuch der Variationsrechnung. 



From the principle of least action we may deduce the equations 

 of motion. Of course the principle was itself derived from these 

 equations, therefore, as is always the case, we obtain by mathematical 

 transformations no new facts. It is however instructive to see how 

 by assuming the principle of least action as a general principle we 

 may obtain the equations from it. 



Let us put in equation 12) 



fl<3 /7/V.2 I f]3 l ,7*2 



t*o r (*dj r -J- ll'i/r \ W6 r , 



*~ - < - = y 



dq ~ dq dq 



giving 



17) 

 If we put 



since P involves all the coordinates and velocities x r , y r> z r , x' r , yl-, 0' n 



