SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 163 



We easily see that the equation of a bicircular quartic having proper- 

 ties (i) and (m) is of the form (24). Hence we may state 



Theorem 2. The ooi trajectories passing through a given point in a 

 given direction have associated with them their orthogonal projections in 

 the tangent plane to the surface at the given point. The loctis of the foci 

 of the osculating parabolas of the associate system is a bicircular quartic 

 having the given point as node and the given direction both as tangent line 

 and also as one of the asymptotes of the hyperbola tchich is the inverse of 

 the quartic with respect to the given point. 



Returning to our bicircuhar quartic (24), we have already noted 

 that the giA'en point is a node and that the two tangents at are 

 given by (26). The first of these has the direction of the initial 

 element v', and the second has the direction ^ given by 



(28) ^ = 



Hence 



(29) 



^2_ _ <Ar'- + 2<^/-i// 

 Az~ (t>v'~ -2xpv'- 4>' 



xP - (pv' _ /- ^ 



I ' 



or 



(30) tan^i = tan 02 



where Q\ is the angle between the initial direction v' and the direction 

 of the force vector \^/0, and 02 is the angle between the direction of the 

 second tangent ^ and the initial direction v' . Hence we have 



Theorem 3. The focal locus described in Theorem 1 has two distinct 

 tangents at the given point. The initial element, which has the direction 

 of one of these tangents, bisects the angle between the force vector and the 

 second tangent. 



§ 4. Curves with Properties I and II. 



Theorems 2 and 3 express geometric properties of the system of oo^ 

 trajectories on the surface. The question arises whether these proper- 

 ties are characteristic of the system, i.e. whether the system of tra- 

 jectories is the only system enjoying these properties. To answer 

 this, let us now find all the systems of 0°^ curves on a surface which 

 possess 



Property I. The oo^ curves passing through a given point in a 

 given direction have associated with them their orthogonal projections 



