60 
ME. CLERK MAXWELL ON THE DYNAMICAL THEORY OF GASES. 
to deal with, $ occurs under two forms only, namely, sin 2 Q and sin 2 20. If, therefore, we 
can find the values of 
15,=^ A^rbdb sin 2 0, and B^J^ irbdb sin 2 23, (8) 
we can integrate all the expressions with respect to b. 
Bj and B 2 will be functions of Y only, the form of which we can determine only in 
particular cases, after we have found 0 as a function of b and Y. 
Determination of 0 for certain laws of Force. 
Let us assume that the force between the molecules Mj and M 2 is repulsive and varies 
inversely as the nth. power of the distance between them, the value of the moving force 
at distance unity being K, then we find by the equation of central orbits, 
(9) 
where x=^, or the ratio of b to the distance of the molecules at a given time : x is there- 
fore a numerical quantity ; a is also a numerical quantity and is given by the equation 
_,/ V 2 MjM 2 y - 1 nm 
“ - Hk(M 1 + M,)) ^ 
The limits of integration are #=0 and x—x\ where x' is the least positive root of the 
equation 
w 
It is evident that 0 is a function of a and n, and when n is known 0 may be expressed 
as a function of a only. 
Also 
so that if we put 
bdb= 
/k(m,+m 2 : 
V v 2 MjM 2 
udc& 
= I T ‘ 
Lra^asin 2 #, A 2 =l ^rada sin 2 2$, 
( 12 ) 
(13) 
A, and A 2 will be definite numerical quantities which may be ascertained when n is given, 
2 
and Bj and B 2 may be found by multiplying A, and A 2 by V n_1 . 
Before integrating further we have to multiply by V, so that the form in which V 
will enter into the expressions which have to be integratedwith respect to and dN a 
mil be 
n - 5 
V"- 1 . 
It will be shown that we have reason from experiments on the viscosity of gases to 
believe that n= 5. In this case Y will disappear from the expressions of the form (3), 
and they will be capable of immediate integration with respect to and <?N 2 . 
