PROPERTIES OF MATTER IN THE GASEOUS STATE. 
795 
Fundamental Theorems. 
80. On the assumptions I. to V., remembering the fact pointed out in Arts. 75 
and 79 with respect to the component velocities of the resultant uniform gas, the 
following theorems are established :— 
Theorem (I .).—Each of the severed inequalities, as defined in Art. 78, in every ele¬ 
mentary group of molecules which in a unit of time leave an element of volume oj 
smcdl hut definite size will severally he less than in the corresponding elementary 
group, which in the same time enter the element in the same direction hy quantities 
which hear approximately the same relation to the mean inequalities of the two 
groups, as the distance through the element in direction of the group hears to a 
distance ( s) which is a function of the density of the gas, and the mean square of the 
velocity of the molecules only. 
To express this theorem algebraically, let Cf and I, as explained in the last article, 
refer to the point in the middle of the element. Then the inequality in the entering 
group is expressed by 
and for the leaving group by 
_dG Sr dl Sr 
dr 2 
dG- Sr , dl Sr 
—. o + i_ G/ r 9 
dr 
And what the theorem asserts is 
J"(G+I)S> —~T.(25) 
wherein s is a function of p and a~ only. 
Proof of Theorem (I.). 
(a) From assumptions I. and II., Art. 79, it follows at once that when the con¬ 
dition of the gas varies from point to point, the molecules cannot enter an element of 
volume in the same manner as they would from any uniform gas. 
(h) From ( a ) and assumption III. it follows that the effect of encounters within an 
element in a varying gas is to render the manner in which the molecules leave as com¬ 
pared with that in which they enter more nearly similar to that of some uniform gas. 
(c) The uniform gas referred to in ( h ) must, as has been already pointed out, have 
component velocities equal to half the mean component velocities of all the molecules 
which in a unit of time pass through the element. 
This at once follows from the illustration appended to assumption III., Art. 79. 
For the molecules which leave an element in a unit of time must have the same mean 
