Voiv. 6, 1920 
MATHEMATICS: J. LIPKA 
621 
troduce an insuperable difficulty regarding specific heat. Even with 
the assumption that I have made as to the length of /, the influence of 
the free electrons, and still more the influence of ionization with increase 
of temperature, on the specific heat, presents a rather serious question. 
Thus, to take what is probably the most unfavorable case set forth in the 
preceding tables, if in Mg q = 1.5 and s = 8 .5 and X'o = 460 R and 
7 = 12%, we have, as the heat absorbed by the free electrons in the rise 
of 1 gm. of Mg from 0 °C. to 1° C, 0.025 cal., while the heat required by 
the accompanying ionization is 0.124 cal., a total of 0.149 cal., which is 
rather more than half the total specific heat of Mg at 0° C. It would be 
easy, howxver, to choose values of q, s and X'o for Mg which would serve 
the purpose of this paper while affecting the specific heat less. More- 
over, it is possible that / may be greater than I have taken it to be. 
Table 17 of my paper in these Proceedings for March, 1920, now re- 
quires revision, Al, Fe, Mo and Tl going into Section (A) ; but the general 
testimony of the table remains unchanged, the mean of the last column 
being now -1.50% for Section (A) and +2.47% for Section (B). 
It remains to be seen whether the theory I am developing can deal 
successfully with the Peltier effect. A preliminary examination, already 
made, of this matter is encouraging. 
^ Kquipartition of energy, as holding for electrons within an atom, cannot be regarded 
as an established principle. 
2 Physic. Rev., Ithaca, April, 1918 (329). 
^ In my previous paper 5o had a different meaning. 
^London, Phil. Mag., 28, 1914 (692-702). 
5 Physic Rev., Ithaca, May 1919 (374-391). 
MOTION ON A SURFACE FOR ANY POSITIONAL FIELD 
OF FORCE 
By Joseph Lipka 
Department of Mathematics, Massachusetts Institute of Technology 
Communicated by Edwin B. Wilson, July 31, 1920 
1. The first part of the paper presents a study of the geometric proper- 
ties of the system of trajectories generated by the motion of a particle 
on any constraining surface (spread of two dimensions) under any posi- 
tional field of force. The complete characteristic properties are derived.^ 
Starting at any point on the surface in a given direction, and with a given 
speed, a unique trajectory is generated. The complete system of tra- 
jectories forms a triply infinite system of curves corresponding uniquely 
to a g^iven field of force. The five geometric properties stated in section 2 
are characteristic of the system of trajectories, and any triply infinite 
system of curves on a surface possessing these five properties may be con- 
sidered as generated by the motion of a particle in a unique field of foroii. 
