CHARACTERISTIC FUNCTION TO SPECIAL CASES OF CONSTRAINT. 149 

 8t 1 dx dr (\ dx s 



or _ 1 dx Or /l dx\ 



"o 

 &c. &c. 



and §!_=- f *• 



x o> Voi z o 



/ 



5. Hence, if t could be found as a function of z,y,z,% ,y ,z , and H, it is 

 obvious that its partial differential coefficients with respect to these quantities 

 would give the motion completely. 



But, neglecting altogether the initial limit, we see that 



/dr\ 2 /(?t\ 2 /-^t\ 2 _ 1 f/dx\ 2 /<%\ 2 , (dz > \\ 

 \dlc) + \dy) + \fa) ~ v* \\dt) + \dt) + \dt) ) 



1 1 



~y 2 ~2(H-V) • ' • ( 2 )- 



6. It can be easily shown, by a process similar to that employed for Varying 

 Action* that, if any integral of this equation can be found, its partial differential 

 coefficients with respect to x, y, z are respectively equal to the corresponding 

 components of the velocity, in a curve which is a brachistochrone for the given 

 forces, each divided by the square of the whole velocity. 



A complete integral of (2) must of course contain, besides H, two arbitrary con- 

 stants a, (3. If, then, t be a complete integral, the equations of the brachisto- 

 chrone are easily shown to be 



^= a - U=» (3); 



where & and 3$ are two new arbitrary constants. 

 Also we have the relation 



dr 



dR 



/'dt Cds //4 v 



7. Before proceeding farther with the theory, we may apply the results 

 already obtained to one or two well-known problems ; commencing with the 

 original case proposed by Bernoulli. 



8. To find the brachistochrone, when gravity is the only impressed force, and the 

 particle has the velocity due to a fall from a given horizontal plane. 



Taking the axis of y vertically downwards, we have 



V= - gy. 

 Also, we may write 



K = ga. 



* Thomson and Tait's Natural Philosophy, § 323, or Tait and Steele's Dynamics of a 

 Particle (2d edition), §§ 252, 253. 



