CHARACTERISTIC FUNCTION TO SPECIAL CASES OF CONSTRAINT, 

 while the equation of the free path is 



r r \ n-2 n _2 



/r\t=l n-2 , a , n. 



The above integration fails in the case of n = —2 ; that is, when the force is 

 repulsive and directly as the distance, the velocity vanishing at the centre of 

 force. But in this case 



t = ad + J- - a 2 log Or, 

 r 



and the equation of the brachistochrone is 



a 



«=0- /J~~ 2 logCr, 



^ fJL a 



the logarithmic spiral. Eliminating r between these equations, we see that the 

 time is proportional to the polar angle. 



Since a definite form has been assigned to the expression for the velocity in 



dr 



this problem, it is obvious that H is given, and therefore that there is no -r^- 



The assumption 



^ = 

 d(p 



is easily justified, in the case of any equation of the form 



^Y-uiY^Y-L * f dT Y--F2 



\dr) + r 2 \dd) + r 2 sin 2 6 \d<[>) ~ ' 



if F be a function of r only. For 



o,/cZt\ d 2 r «, d' 2 r * n d 2 r «, 



8 W) = drift 8r + ddd4> 86 + W 8 * ■ 

 But 



dr _ ™ dr dr _ -^ 2 rdd dr _ 2 r smdd(p 



dr ~~ dt ' rdd ~ dt ' r sinBdcf) ~ dt 



Hence 



* /dr\ _ 8t_ jdr_ d 2 T 1_ dr d 2 r 1 (fr (Pt\ _ St fd¥\ 



\d(p) ~ F 2 \dr drdcf) r 2 d6 dddQ + r 2 sin 2 d d(p d^jF V#j ~ 



dr 



That is, unless F contains <p, t-t is necessarily a constant, (3 suppose. 



But, in the present case, if we give this constant any value but zero, we intro- 

 duce a problem much more general than that proposed, for the expression for the 

 reciprocal of the square of the velocity becomes 



r!_ ft 2 



fj. r 2 sin 2 <9 ' 



