236 
PROFESSOR CLERK MAXWELL OX STRESSES IX RARIFIED 
£i=£i- 
M, 
Mi + Mo 
((£ —£i) 2 sin2 0 +l(v- 2 — ViY+iCz— £i) 3 ]*sin 26 cos <f>) 
( 1 ) 
-When the two molecules are of the same kind, 
M, 
M 1 + M 3 ~ 
and in the present 
investigation of a single gas we shall assume this to be the case. 
If we use the symbol 8 to indicate the increment of any quantity due to an 
encounter, and if we remember that all values of (f) are equally probable, so that the 
average value of cos </j and of cos 3 <£ is zero, and that of cos 3 6 is we find 
% + &) — 0 .( 2 ) 
a(^ 3 +4 3 ) = -[3(4-^) 2 -Y 3 ]sin 3 ^cos 2 0 .(3) 
-f(^ + a[3(^-^) 2 ~V 2 ] sin 2 0 cos 2 0 .(4) 
From these by transformation of coordinates we find 
^(^i 1 7i + ^ 2 ' 1 ? 2 ) = —3 (4 — X^a—^i) sin 2 0 cos 3 d.(5) 
S (£fW + &% 2 ) = — i[9 (liW + tihi) ~ 3 (£iW + £ffh 2 ) 
-(^ + ^)(6^ 2 +y 3 )]sm 2 ^cos 3 0.(6) 
S(£r7i£i + = ~ M. 9 (£iVi£i + — 3(£p?i4 -f + 
+ dzVil# + &7a£i)] sin 3 0 cos 2 0.(7) 
[.Application of Spherical Harmonics to the Theory of Gases. 
If we suppose the direction of the velocity of M, relative to M 2 to be indicated by 
the position of a point P on a sphere, which we may call the sphere of reference, then 
the direction of the relative velocity after the encounter will be indicated by a point 
P', the angular distance PP' being 29, so that the point P' lies in a small circle, every 
position in which is equally probable. 
We have to calculate the effect of an encounter upon certain functions of the six 
velocity-components of the two molecules. These six quantities may be expressed in 
terms of the three velocity-components of the centre of mass of the two molecules (say 
u, v, w), the relative velocity of M 1 with respect to M 2 which we call V, and the two 
angular coordinates which indicate the direction of V. During the encounter, the 
quantities u, v, w, and V remain the same, but the angular coordinates are altered 
from those of P to those of P' on the sphere of reference. 
Whatever he the form of the function of f, rp, r). 2 , £ 3 , we may consider it 
expressed in the form of a series of spherical harmonics of the angular coordinates, 
their coefficients being functions of u, v, w, V, and we have only to determine the effect 
of the encounter upon the value of the spherical harmonics, for their coefficients are 
not changed. 
