160 LIPKA. 



the a's and /3's being functions of (p, \p, and X, and hence of u and v. 

 In deriving the geometric properties of the trajectories we shall use 

 the equation in the form (11) almost exclusively. 



The triply infinite system of curves represented by (11) is thus 

 uniquely determined by the force components 0, ^. Two different 

 fields of force <^, ^p and </>, -^ cannot give rise to the same system of 

 trajectories. For if the two systems 



(14) G' = G{a + hv' + c/ 2 + d'c") and 6" = G'( a + W + ~cv' '' + dv") 



coincide, we must have 



(15) a = a, b = h, c =~c, d = d. 



The last of these equations, by comparison with (12) and (13), gives 



(16) 4^/^ = 0/0, 

 so that we may write 



(17) xf = mp, ^ = acf), 



where a is some function of m, v; substituting these values in the first 

 three equations, we find 



(18) a„ = 0, ttj, = 0, 



and hence a is simply a constant, k. Therefore the forces are the 

 same except for a constant factor, which corresponds merely to a 

 change in the unit of force. Hence we may state 



Theorem 1. The system of trajectories corresponding to a field of 

 force completely defines that field. 



We further note that 6' = satisfies equation (11), so that the 

 geodesies form part of every system of trajectories, i.e. for every field 

 of force. Indeed, from (8), we note that the speed of the particle 

 describing a geodesic is infinite. 



§3. The Associate System and the Focal Locus. 



Consider the oo^^ trajectories passing through a point in a direction 

 v', i.e. having a common initial element {u, v, v'). Project these 

 curves orthogonally into the tangent plane at 0.* We shall call the 



4 For our purpose it is merely necessary to project the elements up to the 

 third order, v', v", v'". 



