PROPERTIES OF MATTER IN THE GASEOUS STATE. 
783 
u, v, w; and owing to the fact that in all the cases to he considered U 3 , V 3 , W 3 are 
of the second order of small quantities compared with u 2 , v'\ id 2 this may be done. 
For we may put 
u+ u+ |+ m+ and <U 
<r*(Q)=2{(£+U)Q}=2{(f+U)Q}+2{(f+U)Q} .... ( 12 ) 
u — u— (— u+ and <U 
<r.,(Q)=2{(f+U)QS = S{(f+U)QJ-S{(f+U)Q} .... (13) 
and when U is small compared with the last term on the right in each of these 
equations will be small to the second order as compared with the. first term. For the 
number of molecules over which the summation in these terms extends is to the whole 
number of molecules in a unit of volume in something less than the ratio of U to 
Hence, as will subsequently appear, in neglecting these last terms we shall be neglect¬ 
ing nothing within the limits of our approximation. We have therefore 
<Ur(Q)=2 {(£ +U) Q} 
f _ > .(14) 
MQ)=^{(I+U)Q} - 
and similarly for all other groups. Thus it appears that the letters a, b, c, &c., may 
be used indifferently to indicate the groups as distinguished by the signs of u, v, tv 
or of r), C 
Distribution of velocities amongst the molecules. 
72. Although not actually essential to this investigation, as it will tend greatly to 
simplify the results obtained, I shall adopt the conclusion arrived at by Professor 
Maxwell* with respect to the distribution of velocities amongst the molecules of a 
uniform gas, viz.:— 
-vr _ r-+V^+C- 
a= d^dridti, .. (15) 
a 7T 3 ' 
where N is the whole number of molecules in a unit of volume, and c/N the number 
whose component velocities lie between £ and £-\-df, rj and 17 -\-drj, and £ and £+</£. 
From equation (15) we have for a uniform gas 
a 
?= 
(1«) 
MDCCCI.X XIX. 
* Phil. Trans., 18G7, p. 65. 
5 H 
