622 
MATHEMATICS: J. LIPKA 
Proc. N. a. S 
An additional property serves to characterize the motion when the field 
of force is conservative. 
Another part of the paper presents briefly an analogous study for cer- 
tain other classes of triply infinite systems of curves on a surface, in par- 
ticular, brachistochrones and catenaries in a conservative field of force. 
For all such systems five characteristic properties (sec. 3) differing but 
slightly from those for trajectories are derived. 
2. The motion of a particle on a surface 
X = x{u,v), y — yiu,v), z — z{u,v) 
may be most simply expressed by the Lagrangian equations 
dt\du ) du ' dt\ dv J dv ' 
where T is the kinetic energy, and 0, \l/ are the components of the force 
given as functions of the coordinates^ w,?;. The differential equation of 
such a system found by eliminating the time has the form 
v"' = A -i- Bv' 4- Cv'\ 
where A,B,C are special functions of u,v,v'. 
The following five characteristic properties of dynamical trajectories 
under any positional field of force are derived by studying the differential 
equation of such a system. 
I. The 00 ^ curves passing through a given point O in a given direction ^ 
haVe associated with them their orthogonal projections in the tangent 
plane to the surface at 0. The locus of the foci of the osculating para- 
bolas of the associate system is a bicircular quartic with O as node, and 
the straight line in the direction ^ both as tangent line to the quartic and 
also as one of the asymptotes to the hyperbola which is the inverse of the 
quartic with respect to the given point 0. 
II. The focal locus or bicircular quartic associated by Property I with 
each direction ^ through a point 0, is such that the tangent line to the 
quartic in the direction ^ bisects the angle between the force vector 
through 0 and the second tangent to the quartic. 
III. Through every point 0 on the surface and in every direction ^ 
through that point, there passes one curve of the system which hyperoscu- 
lates its corresponding geodesic circle of curvature. The locus of the cen- 
ters of geodesic curvature of the oo ^ hyperosculating trajectories which 
pass through 0 is a conic passing through 0 in the direction of the force 
vector. 
IV. The points of the surface and the directions of the force vectors 
through these form a set of differential elements defining a simple system 
of 00^ curves on the surface, viz., the lines of force. At any point O, 
the geodesic curvature of the line of force through 0 is equal to three times 
the geodesic curvature of the hyperosculating trajectory of Property III 
which passes through O in the same direction. 
