CHAKACTERISTIC FUNCTION TO SPECIAL CASES OF CONSTRAINT. 165 



at once to two conclusions : b is an absolute constant ; and so is 2ga, for which 

 we may write c. a is therefore inversely as the diameter of the circle ; and 



-,2 



+ r 2 



From the form of the equation of the path it is obvious that — b 2 is the 

 rectangle under the segments of any chord drawn through the centre of force. 



Hence, in the optical problem, if a ray leave, in any direction, a point distant 



r from the origin, it will pass through another point in the prolongation of r, 



b 2 

 distant — from the origin ; and, in the kinetic problem, there is an infinite number 



of brachistochrones (circles all, and the time being the same for all) when two 

 points thus related are taken as the initial and final points. 



25. Such examples might be multiplied indefinitely. For instance, if the 

 refractive index of a medium be inversely proportional to the square root of the 

 distance from a given point, the path is a parabola about the point as focus ; 

 that every ray may be a cardioid whose cusp is at the point, the square of the 

 refractive index must be inversely as the cube of the distance : and so on. 



26. The processes of § 4 may of course be applied to innumerable problems 

 besides the determination of the form and properties of brachistochrones, but I 

 shall content myself with an example or two. Thus, if we take 



as the characteristic function, we have 



d<P f( v ) dx . , dO r 



d^= vdt> &c -' and m=Jf^ dt 



Of this, besides the cases f(y) — v, and /(#) = -, which we have already con- 

 sidered, the most curious is that where 



V 2 



/M = |; 

 that is, when the space average of the kinetic energy is a minimum. In this case. 

 /<20\ 2 (d&\ 2 (d&\ v * 



, d<£> 



and -tj = s. 



dH 



Again, if we take 



O =J*F(x,y,z)f(v)ds 



d$> -Ffdxn A d<& r 



-dx-=vdt> &C -^ n<i dll=J F SM dt 



