THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 281 
Boltzmann’s equation, viz., by the use of the aggregate of the equations of transfer 
for certain infinite sequences of functions of the molecular velocities, an expression for 
the velocity-distribution function similar to that found by Enskog can be obtained, 
and general formulae for the coefficients can be determined. The present paper contains 
the solution for a gas in which the mean velocity and the temperature vary from 
point to point, the results being worked out at all completely only for the case of a 
simple gas ; in a later paper I hope to give the solution in the most general terms, 
so as to yield a complete theory of viscosity, thermal conduction, and diffusion in a 
composite gas formed of two kinds of spherically symmetrical molecules of any type. 
The formulae obtained by the present method lend themselves to numerical calcula¬ 
tion, and are found to converge rapidly. The results for any particular molecular 
model can be calculated to any desired degree of accuracy ; in this paper three special 
types of molecule have been considered, viz., point centres of force varying inversely 
as the n th power of the distance, rigid elastic spheres, and rigid elastic attracting 
spheres. It is found that, for such molecules, the errors in the approximate formulae 
for viscosity and thermal conduction which were given in my first paper do not exceed 
two or three per cent, at most. The detailed numerical results, and comparison with 
observed data, are given in §§ 10-12. 
§ 2. Definition and Preliminary Consideration of the Problem. 
The Nature of the Gas. 
§ 2 (A). The gas contemplated in our calculations is monatomic and nearly perfect, 
“ monatomic ” implying nothing more than spherical symmetry of the molecules, 
while “ nearly perfect” denotes a certain state as regards density and temperature ; 
this state is such that the molecular paths are sensibly rectilinear for the majority 
of the time, being altered by mutual encounters, the duration of which is a very 
small fraction of the average interval between two encounters. In these circum¬ 
stances the number and effect of encounters in which more than two molecules are 
simultaneously engaged is negligible in comparison with the number and eflect 
of binary encounters. 
The gas is supposed to be acted upon by external forces, and the variations of these 
forces, and of the density, mean velocity, and temperature of the gas, with regard 
to space and time, are small quantities of the first order at most. In the present 
paper the density of the gas is • supposed such that the mean length of path of 
a molecule between collisions is small compared with the scale of the space-variation 
of the above quantities ; the modifications of the theory in the case of highly rarefied 
gases, where the mean free path becomes large, will be dealt with in a future 
paper. As we are not interested in the mass motion or acceleration of the gas 
as a whole, but only in the small variations with regard to space and time, it is 
convenient to imagine that, by the addition of a suitable uniform motion and field 
2 Q 2 
