174 
MATHEMATICS: H. B. PHILLIPS 
Proc. N. A. S. 
From (4) its momentum parallel to the #-axis due to the motion of the 
liquid at that point will be m w t. 
From (6) the total ^-component of momentum transferred in the positive 
direction across unit area of the rry-plane in unit time will then be 
~ oo 
m wHf(w) dw. 
JL f°° 
V — 8 J o 
There will be an equal transfer due to molecules moving in the negative 
direction. Hence 
v f 00 
7) = — I 2m w 2 tf{w)dw. (9) 
V — 8 J o 
The time t can be considered as the interval between collisions at times 4 
to, t%. Since the molecule is assumed to move with constant velocity 
between collisions, 
2mwH = 2 P 1 mw 2 dt = nh, (10) 
J to 
where, according to Sommerfeld's theory, n is an integer. Also 
* fiw) dw=N/2M i (11) 
0 
since it is the number of molecules per gram for which w is positive. Com- 
bining (9), (10), and (11), we get (1) which was to be proved. 
3. Owing to lack of data on the equation of state, the only substances 
on which the formula can at present be tested are carbon dioxide, ether, 
and mercury. Using the values of N and h given by Birge, 5 
3iV7* = 3(6.0594)(6.5543)10- 4 =. 011914, 
and so equation (3) takes the form 
»7(fl-s) = .011914/M. (12) 
Values of 5 for carbon dioxide and ether, 6 tables I and II, were supplied 
by Professor Keyes. At low temperatures the measured viscosities and 
those calculated by equation (12) do not differ by more than the experi- 
mental error. In case of carbon dioxide the temperature 30° is too near 
the critical point (£ = 31°) for satisfactory use of the equation of state. 
Above 10° the calculated viscosity of ether is too large, the difference in- 
creasing with the temperature. This may be due to the fact that ether is 
a complex of more than one type of molecule. The equation of state was 
determined on the assumption that each liquid molecule is formed by the 
combination of two gas molecules. This may be substantially true at 
10° and not at 100°. 
To obtain values oi v — 8 for mercury, I make use of the fact that mon- 
atomic substances seem to have constant co-volumes 7 8. Since mercury 
is monatomic in the gas phase, I assume that 8 is constant or nearly con- 
stant in the liquid phase. Also the term 
RT/(v-8) 
in case of mercury is very large (more than 15000 atmospheres). Hence 
at atmospheric pressures we may neglect p and so write (2) in the form 
