Vol.. 7, 1921 
MATHEMATICS: N. WIENER 
297 
ity that the total momentum acquired by a particle from the impacts of 
molecules will lie between x 0 and %\ is of the form 
1 /•&_«? 
V -n-ctJ a 6 * d% 
Superimposed on this momentum is that due to the viscosity acting in ac- 
cordance with Stokes' law, namely QttttjV, where V is the velocity of the 
particle. We shall write Q for Qirrr] -r-M, where M is the mass of a particle. 
Let us write r for ct. Let the total impulse received by a given particle 
in time t, neglecting the action of viscosity, be / (r). Consider m (t), the 
actual momentum of the particle, as a function of t. Then 
m(t + dt) = m(t) + f(ct + cdt) - f(ct) - Qm{ct + cddt)dt (o <&<_!). 
We cannot treat this as a differential equation, as we have no reason to sup- 
pose that / has a derivative. We can make it into an integral equation, 
however, which will read 
w(0 - m{o) = f(ct) - Qflmifydt. 
Clearly one solution of this integral equation is 
m(t) = m(o)e- Qt + f(ct) - Qe~ Qt f*f(ct)e Qt dt, 
and there is no difficulty in showing that an integral equation of this sort 
can have only one continuous solution. 
Another integration gives for the distance traversed by the particle in 
time t 
= m^Cq a ~ e ~ QI) +e ~°'fo eQ%ct)di ) (2) 
Applying the methods of my previous paper, we get for the mean value of 
d 2 h in accordance with (1) ; 
— 1 r m(o) I 2 1 r ct r ct \ Q(* + y) 
