Voi,. 7, 1921 
MATHEMATICS: H. B. PHILLIPS 
173 
the velocity at the point (x,y,z), 
bu 0 /bz = l. (4) 
Let the components of velocity of a molecule parallel to the x and z axes be 
u, w respectively. When a molecule of mass m moves in the positive di- 
rection of the 2-axis across the ^-plane, the ^-component of momentum 
transferred across that plane is mu. When it crosses in the negative direc- 
tion the transfer is — mu. Under these conditions the viscosity r\ of the 
liquid is defined as the total ^-component of momentum transferred across 
unit area of the xy -plane in unit time. 2 To find the viscosity we there- 
fore find the number of molecules of each velocity crossing unit area and 
multiply by the average momentum transferred by each. 
Let / (w) dw be the number of molecules per cubic centimeter with com- 
ponents of velocity parallel to the 0-axis between w and w + dw. If the 
molecules did not influence each other, the number of these that would 
cross unit area of the ^-plane in unit time would be 
w f(w) dw. (5) 
Because of the interference of molecules with each other, this must be re- 
placed by 
v 
— : w f (w) dw. (6) 
v—o 
To see this we note that if the molecules moved independently, the z- 
component of momentum transferred across unit area of the rry-plane in 
both directions in unit time would be the pressure 
RT/v. (7) 
From equation (2), in the real liquid this is replaced by 
RT/(v-5) (8) 
Now the mass of the molecule and the temperature (average kinetic energy) 
are the same in both cases. Hence the increased momentum (8) com- 
pared with (7), can only be due to an increase in the number of molecules cross- 
ing in unit time. Since the same law of distribution of velocities (Maxwell's 
Law) is assumed in both cases, the ratio in which the numbers are increased 
must be the same v/(v— 8) for all velocities. Thus the number of mole- 
cules with velocity components between w and w + dw crossing unit area 
of the xy-plane in unit time is given by equation (6). 
The velocity u of a given molecule parallel to the #-axis will not always 
be equal to the average velocity u 0 of the liquid at that point. There will, 
however, be a series of instants (which we may call collisions) when u will 
be equal to u 0 . I assume that the interval between two such instants will 
be a half period (interval between libration limits) 3 in the sense of the 
quantum theory, and that the molecules can be. treated as moving with 
constant velocity between consecutive collisions. Suppose then a mole- 
cule reaching the xy-plane has been moving a time, t, since its last collision. 
