172- 
MATHEMATICS: H. B. PHILLIPS Proc.N. A. S. 
"spin faster," i.e., increasing the velocity of the circulation, decreasing 
the time required for a given mass to complete the cycle. In either case 
he has increased the energy turn-over per unit of time. Whether, in this 
he has been unconsciously fulfilling one of those laws of nature according 
to which certain quantities tend toward a maximum, is a question well de- 
serving of our attention. 
* Papers from the Department of Biometry and Vital Statistics, School of Hygiene 
and Public Health, Johns Hopkins University, No. 44. 
!Lotka,A. J., Physic. Rev., 24, 1912 (235-238); /. Washington Acad. ScL, 2, 1912 
(2, 49, 66); Science Progress, 55, 1920 (406-417); Proc. Am. Acad. Arts ScL, 55, 1920 
(237-153); these Proceedings, Sci. 6, 1910 (410-415). 
2 Proc. Amer. Acad., loc cit., p. 142. 
3 See for example Picard, Traite d' Analyse; H. Bateman, Differential Equations, 
1918, p. 245. 
4 Spencer, First Principles, Chapter XXII; Winiarskie, "Bssai sur la Mecanique 
Sociale," Revue Philosophique, 44, 1900 (113). 
6 At the time of reading proof this project is partially realized. A discussion of the 
applicability of the Le Chatelier principle to systems of the general character here 
considered will appear in a forthcoming issue of the Proceedings of the American 
Academy of Arts and Sciences 
6 I<otka, A. J., London, Phil. Mag., Aug., 1911, p. 353. 
A FORMULA FOR THE VISCOSITY OF LIQUIDS 1 
By H. B. Philips 
Department oe Mathematics, Massachusetts Institute oe Technology 
Communicated by A. G. Webster, March 22, 1921 
1. In this paper I obtain for the viscosity of a liquid the formula 
nN h 
v ~ 2M («-*)' (1) 
where N is the number of molecules in a mol, h is Planck's constant, M is 
the molecular weight of the liquid in the gas phase, v its volume per gram, 
and n an integer. The quantity 5 is the co-volume as used in the equa- 
tion of state of Keyes 1 
RT__A_, (2) 
V V-8 {v-D 2 K } 
In all the cases to which I have applied the formula, n = 6 and so (1) takes 
the form 
v (v-8) = SNh/M (3) 
It is to be noted that SN/M is the number of translational degrees of 
freedom of the molecules in the volume v of the liquid. 
2. To prove equation (1), let x,y,z be rectangular coordinates and sup- 
pose the liquid to flow parallel to the #-axis in such a way that, u 0 being 
