116 G. F. Becker — Maximum Dissipativity. 



The last term in this equation is infinitesimal compared with 

 the other terms and may be neglected. Hence 



pdS _ ds Pj 

 p,d$ ~" c?Sj " p ' 



Let x and x 1 be arcs of the curves on which the action is the 

 same, so that xu=x 1 v i . Then since in general v 1 dt=ds 1 and vdt 

 = ds, the velocity being represented by v, 



Kj v ds _ Pj 

 h v t c/s, p ' 



That is to say, the radii of curvature at the extremities of the 

 arc ds, or, more generally, perpendiculars to the two curves at 

 the extremities of this arc, intercept arcs en the two curves on 

 which the action is the same. It evidently, follows that any 

 two lines perpendicular to the two curves intercept arcs on 

 which the action is the same. A more general proposition em- 

 bracing this is one of the immediate deductions from Hamilton's 

 principle of varying action.* 



The action on any natural path is a minimum provided that 

 the path is sufficiently short, but it is clear, from the last para- 

 graph, that perpendiculars to the undisturbed path and the dis- 

 turbed path at their points of intersection cut both paths, and that 

 the action is consequently the same on each between the same 

 initial and final positions. It can, therefore, be a true minimum 

 on neither. The action, consequently, cannot be a minimum 

 for a distance greater than that between these points of inter- 

 section. These points are called conjugate kinetic foci by 

 Messrs. Thomson and Tait, who give propositions embracing 

 those just stated, but reached by a different method ; and who 

 also show that while the action from a given configuration up 

 to the first kinetic focus is always a minimum, it may cease to 

 be the least possible before a kinetic focus is reached. f The 

 variation of the action always vanishes, however, and when the 

 action ceases to be a minimum it must become a maximum or 

 a minimax. 



If the path which a particle pursues returns upon itself and 

 if the motion is stable, a kinetic focus conjugate to the initial 

 point usually occurs at the completion of the circuit, coinciding 

 in position with the starting point. Even when this is not the 

 case, however, the action for an entire circuit is the same on 

 the disturbed path as on the undisturbed path. That it is the 

 same up to the last kinetic focus before the circuit is complete 

 is evident. The starting point may be arbitrarily chosen and 

 may therefore be taken at a point where the movement is per- 

 pendicular to the line of force. This line then cuts the dis- 



* Thomson and Tait, Nat. Phil., § 332. \ Nat. Phil., § 358, et seq. 



