518 
PROFESSOR J. J. THOMSON ON SOME APPLICATIONS OF 
system whose motion is to explain the frictional forces, then we have, by Lagrange’s 
equation, 
d dT dT d dT dT , dV 
—-r — —-- —= - - —p — = external iorce ot type x ; 
dt dx dx dt dx dx dx 
thus the term 
ddT_cJT 
dt dx dx 
must be equal to the “frictional term” which is proportional to x. For this to be 
the case, it is evident that T' must involve x. The momentum of the system is, 
however, d (T -|- T ")/dx. This momentum must, however, be the same as that given by 
the ordinary expression in Rigid Dynamics, viz., dT/dx. If these two expressions, 
however, are identical, dT'/dx must vanish for all values of x, that is, T' cannot involve 
x', which is inconsistent with the condition necessary in order that the motion of 
the subsidiary system should give rise to the “ frictional ” terms. Hence we conclude 
that the frictional terms cannot be explained by supposing that any subsidiary system 
with a finite number of degrees of freedom is in connection with the original system. 
If we investigate the case of a vibrating piston in connection with an unlimited 
volume of ah-, we shall find that the waves starting from the piston dissipate its energy 
just as if it were resisted by a frictional force proportional to its velocity; this, 
however, is only the case when the medium surrounding the piston is unlimited ; when 
it is bounded by fixed obstacles the waves originated by the piston get reflected from the 
boundary, and thus the energy which went from the piston to the ah gets back again 
from the air to the piston. Thus the frictional terms cannot be explained by the 
dissipation of the energy by waves starting from the system and propagated through 
a medium surrounding it, for in this case it would be possible for energy to flow from 
the subsidiary into the original system, while, if the frictional terms are to be explained 
by a subsidiary system in connection with the original one, the connection must be 
such that energy can flow from the original into the subsidiary system, but not from 
the subsidiary into the original. 
Hence we conclude that the equations of motion, with frictional terms in, represent 
the average motion of the system, but not the motion at any particular instant. 
Thus, to take an example, let us suppose that we have a body moving rapidly 
through a gas; then, since the body loses by its impacts with the molecules of a gas 
more momentum than it gains from them, it will be constantly losing momentum, and 
this might on the average be represented by the introduction of a term expressing a 
resistance varying as some power of the velocity; but the equations of motion, with 
this term in, would not be true at any instant, neither when the body was striking 
against a molecule of the gas, nor when it was moving freely and not in collision with 
any of the molecules. Again, if we take the resistance to motion in a gas which arises 
from its own viscosity, the kinetic theory of gases shows that the equations of motion 
