498 REPORT — 1889. 



2. On some Formulae connected with Bessel's Functions. By Dr. Meissel. 



3. On the Belations hetween Bay-Curvatures, Brachistochrones, and Free 

 Paths. By Professor J. D. Everett, F.B.S. 



In a paper printed in last year's Report (p. 581) I called attention to the fact 

 that a curve which is a free path for a particle under any given law of force is 

 identical with the path of a ray in a medium in which the index jj. at each point is 

 proportional to the velocity of the particle. This proportionality must be under- 

 stood to hold not only at points lying- on the orbit itself, but at all neighbouring 

 points, the velocity at a point not on the orbit being interpreted to mean the 

 velocity which the particle would have if guided to this point by frictionless con- 

 straint. 



On the other hand, the path of a ray is a path of least time, and would there- 

 fore be a brachistochrone for a particle subject to forces which, with proper initial 

 Yelocity, would cause the velocity at each point to be directly as the velocity of 

 Ijcrlit —in other words, inversely as fi. Thus every curve that would be a ray- 

 path in a medium in which fi is an assigned continuous function of the co-ordi- 

 nates would also be a free path of a particle for one law of force, and a brachisto- 

 chrone for another. Also, if we find a second ray-path by giving fi a value 

 proportional to the reciprocal of its previous value, this second curve will be a 

 brachistochrone for the first law of force and a free path for the second. The two 

 curves thus mutually related may be called ' conjugate.' 



When /i is a function of distance from a fixed centre, the path of a ray is deter- 

 mined by the equation /x;) = C, where ^ is the perpendicular from the centre on the 

 tano-ent, and is constant along any one ray. Let fx be proportional to r" , r being 

 the^distance from the centre, and ti any positive exponent. Then by supposing the 

 velocity of a particle to be directly as fx, we find that a force of repulsion from the 

 centre varying as »-2n-i will make every ray a free path if the velocitj^ be that due 

 to fall from the centre, and that a force of attraction to the centre varying 

 inversely as ?•-"+' will make every ray a brachistochrone if the velocity be that due 

 to fall from infinity. The case n=l gives as the conjugate cur\'es the rectangular 

 hyperbola and the equiangular spiral. The cases n = ^, n = ~, n==S give, as one of 

 the conjugate curves, the parabola, the cardioid, and the lemniscate of Bernouilli 

 respectively. 



When /x is a function of the distance y from a fixed plane, the value of fi cos 6 

 is constant along any one ray, 6 denoting the angle whose tangent is 



di/ 



dx' 

 Denoting this constant value by C, we have therefore 



(l)"=(-o)'-^ 



as the differential equation of a ray. Then, by supposing /i to vary first directly 

 and then inversely as ?/" . we get a repulsive force varying directly as ySu-i ^jth 

 velocity due to fall from the fixed plane, and attraction inversely as j/^i+i with 

 velocity from infinity. The difierential equations of the conjugate curves will be 



m-w- 



and 



/-7,.\ 2 /_ \ 2n 



■1. 



(£y=(r-^ 



The value m = 1 gives as conjugate curves the catenary and the circle, the directrix 

 of the catenary and the centre of the circle bemg in the fixed plane. The case w = ^ 

 gives the parabola and the cycloid, and is applicable to ordinary terrestrial gravity, 

 since it makes the repulsive force constant. 



Again, the curvature of both free paths and brachistochrones must follow the 



