CHARACTERISTIC FUNCTION TO SPECIAL CASES OF CONSTRAINT. 163 



while the time in an elliptic orbit is {of course) measured by the area described 

 about one focus, the action is measured by that described about the other* is easily 

 traced to the fact that the rectangle under the perpendiculars from the foci on 

 any tangent is constant. 



21. It follows from Hamilton's investigations, that in the free ellipse we 

 have 



2 (£ - H) dr 



h 



x/ 2 (^-H)--f 

 \r ) ir 



where a depends upon the excentricity of the ellipse by the formula 



2H 



- ft* a 



The theorem may therefore be generalized as follows : — The free ellipse will be a 

 brachistochrone, if the velocity be given by 



1 



«« = 2(H 1 -V 1 ) = 



2(£-H) |0'(A)} 2 



where <p' is any function, and A is the integral last written. By differentiation 

 with respect to r, we get the law of central force requisite. 



But results of this nature may be deduced to any desired extent, without more 

 trouble than the requisite integrations involve. 



22. The examples immediately preceding are but particular cases of the follow- 

 ing general theorem, which is easily seen to be involved in the results of §§ 16, 19. 

 If we have two curves, P and Q, of which P is a free path, and Q a brachis- 

 tochrone, for a given conservative system of forces ; P mill be a: brachistochrone 

 for a system of forces for which Q is a free path — and the action and time in any 

 arc of either, when it is described freely, are functions of the time and action 

 respectively, in the same arc, when it is a brachistochrone. 



23. It is easy to see, that there exists a very singular analogy between the 

 processes we have just given, and those suggested by certain problems in optics. 



Assuming, for an instant, the exploded corpuscular theory of Light, Varying 

 Action is at once applicable to the determination of the path of a corpuscle. On 

 the other hand, if we assume, as our fundamental hypothesis, that light takes 

 the least possible time to pass from one point of its path to another, the foregoing 

 investigations would be directly applicable to find the path in a medium whose 

 refractive index (on which the velocity depends), at any point, is a given function 

 of the co-ordinates ; in other words, in a heterogeneous singly refracting medium. 



In the beautiful investigations of Hamilton, on the Theory of Systems of Rays 



* Proc. R.S.E. March 1865, or Tait and Steele's Dynamics of a Particle (2d edition) § 258. 

 VOL. XXIV. PART I. 2 Y 



