CHARACTERISTIC FUNCTION TO SPECIAL CASES OF CONSTRAINT. 151 



dr _ _ fds_ _ 1 dr 1_ f dy 



dR J v s ~ g da~ ' W2g*J (a + )2 / 1 _1 



*" a + y b 



2g**a + y 6 "*" a 



~ s/2g 

 The above (at first sight apparently too limited) assumptions 



dx ' dz ' 



and the consequent reduction of the question to a plane problem, may seem to 

 require some justification. This is easily supplied, thus : In the equation 



\dx) + \dy) + \dz) ~ F2 ' 



the direction-cosines of the tangent to the brachistochrone, at the point as, y, z, 

 are, by (1.), 



,_ ldT 1 dT _ 1 dr 



l -¥dx^' m ~F%' n ~Fdz' 



At the adjacent point z + 8x,y + 8y,z + 8z, where we have, of course, 



the value of I becomes 



8x 8y 8z ?> 

 -j- = -* = — = be, 

 I m n 



dr . d 2 r 5, d 2 r 5> d 2 r 



r _ dx dx 2 ' dxdy " dxdz 

 F + SF "~ 



dr 8s (dr d 2 T dr d 2 T dr d 2 r \ 

 _dx F \dx dx 2 dy dxdy dz dxdz J 



F + 8F " 



8s 



dr /d¥\ 

 _ dx \dxj 



~ F + 8F ' 



But in the above problem F is a function of y only, and we must therefore 



have 



V_ = l 



n' n ' 



which shows that the curve is in a plane parallel to the axis of y. 



9. To find the Brachistochrone when the force is central, and proportional 

 to a power of the distance; the velocity being also proportional to a power of the 

 distance, that is, being the velocity from infinity if the force is attractive, from the 

 centre if it is repulsive. 



VOL. XXIV. PART I. 2 T 



