166 PROFESSOR TAIT ON HAMILTON'S CHARACTERISTIC FUNCTION, ETC. 



TT .„ -n / \ Constant 



Hence, if F (a>, y, z) = . . ■ , 



we have -= = Ct, 



so that there is an infinite number of values of the characteristic function, besides 

 that of Hamilton, which give the time through any arc of the orbit by their dif- 

 ferential coefficients with respect to H. 



27. Enough of this ; I conclude with the remark that various investigations 

 in Statics supply us with excellent examples in our subject.* Take the common 

 catenary, for instance, its equation is found by the conditions 



I yds = minimum, and I ds = constant, 



the axis of y being directed vertically upwards. 

 This gives 



Sf(y + a)ds = 0. 



Hence the catenary is the free path of a particle whose velocity is given by 



v = G (y + a) ; 



that is, if the force be in the direction of, and proportional to, the ordinate, and 

 repulsive from the axis of x. In the same way we see that the catenary is the 

 brachistochrone if the velocity be inversely as the distance from the axis ; that 

 is, if the force be attractive, and inversely as the cube of the distance from the 

 axis. 



* Compare Thomson and Tait's Natural Philosophy, §§ 581, 582. 



