TRANSACTIONS OF SECTION A. 615 



Baifcaglini ' determined the components of the forces in the case of motion in a 

 conic; and Dainelli, in the eighteenth volume of Battaglini's ' Journal,' treated 

 the more general one of finding the components of the forces actiiig on the 

 particle as functions of its co-ordinates when the trajectory is a general curve. 

 In a recent number of Crelle's 'Journal,' vol. 131, pp. 136-151, Professor 

 Stephanos has proposed still another generalisation of Bertrand's problem. 

 Bertrand considered the case where the force has, in general, a unique direction 

 at every point and the trajectories are conic sections ; he proved that in this case 

 the force should either pass through a fixed point or remain parallel to a fixed 

 direction. Professor Stephanos examines the general case of conic section 

 trajectories when the force has not necessarily a unique direction at every 

 point ; he finds that this more general problem admits of no other solution than 

 those included in the case considered by Bertrand, and he shows that every force 

 under which a point describes a conic section, whatever be the initial conditions, 

 either always passes through a fixed point or remains parallel to a fixed 

 direction ; fi^nally, by applying his method to the determination of the explicit 

 forms of the forces in the case of conic sections, Stephanos rediscovers the known 

 results due to Darboux and Halphen. 



All these discussions have been concerned with the motion of a single 

 material point. Similar problems may be proposed for material systems. 

 Oppenheim, in the third volume of the ' Publications ' of the Von Kuflher 

 Observatory, investigates the central conservative forces under which three 

 bodies describe given co-planar curves ; he constructs the analytical machinery 

 for the case of three arbitrary orbits, but the resulting diflerential equations 

 ofler insuperable difficulties, and the only particular solutions realised were 

 previously known. . 



The first section of the present paper offers a modest contribution by studymg 

 the problem of Bertrand, in all the generalities considered by Stephanos, but for 

 trajectories which include the conic sections as a very special case. In the second 

 section a wider generalisation is undertaken by proposing to determine the forces 

 capable of maintaining the motion of a particle on an arbitrarily given curve, in 

 space of any dimensions, independently of the initial conditions of the motion. 

 In the third section the forces of a central conservative system, capable of 

 maintaining a system of ??i-particles in motion on as many prescribed, but arbi- 

 trary, orbits in a space of w-dimensions are investigated. In the succeeding 

 sections applications are made to the study of a system of three bodies describing 

 any given tra-jectories in ordinary space, independently of the initial conditions, 

 under central conservative forces. Of special Interest, perhaps, is a system of 

 differential equations which the functions, defining the orbits, must satisfy, if 

 the velocities and accelerations are to be determinate, certain of these equations 

 vanishing identically when the motions take place in a single plane ; and among 

 the results achieved in the plane is the construction of an infinite family, in- 

 volving arbitrary functions of Integrable problems of three bodies, whose force- 

 functions depend only on the masses and the mutual distances of the bodies. 



5. Factorisation of the A.P.F. o/ iV=(2/"q:l). 

 By Lieut. -Colonel Allan Cunningham, B.E. 



Here A.P.F. is short for ' maximum algebraic prime factor.* 

 Let F {n) denote the A.P.F. of N (w) = (y" T 1). 



Then N (105) = F (105) . F (35) . F (21) . F (15) . F (7) . F (5) . F (3) . F (1), 



. f^,^^-^ N (105) . N (7 ) . N (5) . N (:3) 

 whereof F (10o) = j^ (35) . N (2l)TN'(ie)TN (1)' 



the signs T being the same as in N (105) throughout, and is of order y* 



» OiomaU di Matcmatiche, vol, xvii. p. 43 et seq. 



