\'OL. 6, 1920 
MATHEMATICS: J. LIPKA 
623 
V. Construct any isothermal net on the surface. At any point 0, 
this net determines two orthogonal directions in which there pass two 
isothermal curves of the net and two hyperosculating curves of Property 
III. If pi, p2, R\, R2 are the radii of geodesic curvature of these four 
curves, Si, S2, the arc lengths along the isothermal curves, and co, the 
tangent of the angle between the force vector and the isothermal curve 
with arc S2, then, as we move along the surface from O, these quantities 
varv so as to satisfv the relation 
^_ (l) - ^ (I) - ± + -L - ^'(i"g = 0, 
where 
Ki \pi R^/ K2 CO \P2 R2/ 
If, in these five properties, we replace the force vector through a point 
by the tangent line to the conic of property III, we may state the theorem : 
In order that a triply infinite system of curves ( 00 ^ w each direction through 
each point) on a surface may be identified with a system of dynamical tra- 
jectories under any positional field of force, the given system must possess 
properties /, //, IV, V. 
When the surface is a developable surface, the bicircular quartic of 
property I reduces to a circle passing through the given point. 
If the field of force is conservative, the conic of property III reduces 
to a rectangular hyperbola. 
3. If we consider the motion of a particle on a surface in a conservative 
field of force from one position Po to another Pi with the sum of the kinetic 
and potential energies equal to a given constant, h, certain systems of ^ ^ 
curves are defined by 
(Pi) n 
S ^IW -\- h ds = minimum, 
(Po) 
where W is the work function (negative potential) and ds is the element 
of arc length. Among such systems, called "w" systems, we may men- 
tion dynamical trajectories {n = 2), hrachistochrones {n = —2), and 
catenaries (n = 1). We may find the differential equation of the 00 * 
curves of an "7z" system by the methods of the calculus of variation.^ If, 
by differentiation, we eliminate the constant of energy, h, from this equa- 
tion, we shall get the differential equation of the 00 ^ curves of a complete 
system. This equation has the form 
v"' = A Bv" + Cv"\ 
where A, B, C are special function of u, v, v' , and involve n as a parameter- 
The following five characteristic properties of an "n" system on a sur- 
face are derived by studying the differential equation of such a system. 
I'. The system possesses property I of section 2. 
