of Perfectly Elastic Spheres. 31 
a tangential force which constitutes the internal friction of the 
gas. The whole friction between two portions of gas separated 
by a plane surface, depends upon the total action between all the 
layers on the one side of that surface upon all the layers on the 
other side. 
Prop. XIII. To find the internal friction in a system of moving 
particles. 
Let the system be divided into layers parallel to the plane of 
zy, and let the motion of translation of each layer be u in the 
direction of x, and let u=A+Bz. We have to consider the 
- mutual action between the layers on the positive and negative 
sides of the plane zy. Let us first determine the action between 
two layers dz and dz', at distances z and —z! on opposite sides 
of this plane, each unit of area. The number of particles which, 
starting from dz in unit of time, reach a distance between n/ and 
(n+dn)/ is by (19), 
Noe dz dn. 
The number of these which have the ends of their paths in the 
layer dz! is 
v 
Qnk° 
The mean velocity in the direction of x which each of these has 
before impact is A+ Bz, and after impact A+ Bz’; and its mass 
is M, so that a mean momentum = MB(z—2’) is communicated 
by each particle. The whole action due to these collisions is 
therefore 
N —” dz dz' dn. 
NMB ae (z—2!)e-" dz dz! dn. 
We must first integrate with respect to z! between z'=0 and 
z’=z—nl; this gives 
v 
+NMB on (n21?— 2?) e—" dz dn 
for the action between the layer dz and all the layers below the 
plane zy. Then integrate from z=0 to z=nl, 
4MNBlon?e-" dn. 
Integrate from n=0 to n=, and we find the whole friction 
between unit of area above and below the plane to be 
du du 
F=! =lplv— =p— 
3MNLB splv Og 
where yu is the ordinary coefficient of internal friction, 
Mv 
] 
Pate ae aa ws? (24) 
