792 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
motion which differ by less than certain small angles from a given direction, or, in other 
words, those molecules of which the directions of motion are parallel to some line 
which may be included within a pyramidal surface having indefinitely small angles at 
the apex. Such a group will be called an elementary group, and in this sense only 
will the term elementary group be used. The mean ray or axis of the pyramid is the 
mean direction of the group. And it is to be noticed that only those molecules that 
are moving in the same direction parallel to the axis of the pyramid are included in 
the same group, those with opposite motion constituting another elementary group. 
The distinguishing features of an elementary group, apart from the direction of the 
group, are the number of molecules at any instant in a unit of volume—the symbol N 
will be used to signify this number; their mean velocity, mean square of velocity, &c., 
will be indicated without regard to direction by the symbols v, v 2 ; and to avoid 
confusion, instead of using Q to indicate the two latter quantities the letter G will be 
used to represent severally N, v, v 2 , &c. 
The resultant uniform gas. —It has been already pointed out (Art. 75) that if the 
encounters within an element of volume resulted in the molecules leaving the element 
in the same manner as they would leave if the gas about and within the element were 
uniform, this uniform gas must have component velocities which are one-half the mean 
component velocities of all the molecules of the varying gas which in a unit of time 
pass through the element. This uniform gas, which would also have approximately the 
mean pressure and density of the actual gas in the element, is called the resultant 
uniform gas of the gas within the element. U, V, W are used to designate its com- 
px~ 
ponent velocities, p to express its density, and G- to express its pressure. U, V, W are 
functions of u, v, w and of the variations of p, a , or, in other words, they are functions 
of the condition of the gas at the point considered, but they cannot be completely deter¬ 
mined in the first stage of the investigation. 
The inequalities in elementary groups. —All the elementary groups relating to a unit 
of volume in a varying gas are compared with corresponding elementary groups in the 
resultant uniform gas for the element, and the differences in respect of the density and 
velocities of the molecules are spoken of as the inequalities of the group. There are 
only four quantities in respect to which the groups can be compared, namely : the 
numbers of molecules, the mean velocity, the mean square, and the mean cube of the 
velocity; essentially, therefore, the differences in these constitute the inequalities of the 
gorup. 
Thus, if G standing for n, v, v 2 or v 3 refers to an elementary group of the resultant 
uniform gas for an indefinitely small element, and G-f I refers to the corresponding ele¬ 
mentary group of the varying gas, then I represents the inequality in an elementary 
group at a point as compared with the resultant uniform gas at that point. 
When the element has small but definite dimensions (fir) the inequalities of the 
elementary groups entering or leaving will be 
