ME. CLEEK MAXWELL ON THE DYNAMICAL THEOEY OE GASES. 
65 
2nd. The mean square of the velocity is v 2 =^cc 2 (29) 
3rd. The mean value of | 2 is | 2 =i« 2 (30) 
4th. The mean value of is £ 4 =f a* (31) 
5th. The mean value of |V is |V=^a 4 (32) 
6th. When there are two systems of molecules 
M 1 a 2 =M 2 /3 2 , (33) 
whence 
2 vl, (34) 
or the mean vis viva of a molecule will be the same in each system. This is a very 
important result in the theory of gases, and it is independent of the nature of the action 
between the molecules, as are all the other results relating to the final distribution of 
velocities. We shall find that it leads to the law of gases known as that of Equivalent 
Volumes. 
Variation of Functions of the Velocity due to encounters between the Molecules. 
We may now proceed to write down the values of M in the different cases. We shall 
indicate the mean value of any quantity for all the molecules of one kind by placing a 
bar over the symbol which represents that quantity for any particular molecule, but in 
expressions where all such quantities are to be taken at their mean values, we shall, for 
convenience, omit the bar. We shall use the symbols ^ and to indicate the effect 
produced by molecules of the first kind and second kind respectively, and to indicate 
the effect of external forces. We shall also confine ourselves to the case in which w=5, 
since it is not only free from mathematical difficulty, but is the only case which is con- 
sistent with the laws of viscosity of gases. 
In this case V disappears, and we have for the effect of the second system or the first, 
M ]\r / K(M t +M 3 ) \ 
St M t M 2 I 
*aP"(q-q )#, 
(35) 
where the functions of |, jj, £ in J(Q' — Q)d<p must be put equal to their mean values for 
all the molecules, and A, or A 2 must be put for A according as sin 2 Q or sin 2 2$ occurs in 
the expressions in equations (4), (5), (6), (7). We thus obtain 
(«) 
( 0 ) 
Mi_ / 5 Vn M A (2 — 2 ) • 
it ....... 
8 2 £? / K U N 2 M 2 
Vt \M 1 M 2 (M 1 + M 2 ) j m 1 + m 2 
{2A 1 (i 2 -i 1 )(M 1 i i +M 2 | 2 )+A 2 M 2 (^ -2£=1 
K _ V n 2 m 2 
U + M 2 ) J M x + M 2 
{ A, (2M 2 i 2??2 — 2M.U + (M, - M 2 )(U+ U ))- - 3A 2 M 2 (| 2 - 
2.X* 
(36) 
. (37) 
MDCCCLXV1I. 
K 
