THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 291 
where C, U, V, W all have the same suffix 1 or 2, with or without an accent ('). 
Similarly, the mass-centre G has the velocity c 0 or (X„, Y 0 , Z 0 ), and since (by the 
principle of conservation of momentum) this remains invariable throughout the 
encounter, # we have 
(38) c 0 = = [J-iQ'i + ^2 = Mi [ti] + M2[f 2 ] ? 
or 
Co — $'o = [Co]* 
Since, by (38), 
(39) m l {[c,]-[c 0 ]} = -m 2 {[c a ]-[c 0 ]} = \[c^-[_c^} = m 0 (miM 2 )’ /2 [c k ], 
where [c R ] is defined by the equation 
(40) [Ch] = (miM 2 )' 4 {[Ci]-[C 2 ]} ? 
we see that the momentum of the molecules, relative to G, is equal in magnitude but 
opposite in direction in the two cases, its value being ±m 0 (miM 2 ) 1/2 [Cr]- The relative 
velocity of the two molecules is, by (40), equal to (m]M 2 )~ V ~ [Ck] ; this varies throughout 
the encounter, owing to the inter-action of the molecules ; its initial and final values 
are given by 
(41) Cr = (miMs) {Qi g)? Ck = (miMs) "(Ci Q 2)5 
which are special cases of (40). 
Equations (38), (41), and the reciprocal equations 
(42) Cl = C 0 + ^ 21 l -C R C 2 = c u Mi2 ,2 Cj{, 
(43) c\ = c 0 +^ 2x k c' u c' 2 = c 0 — 
indicate that c, c 2 or c\, c' 2 are equivalent to c 0 , c R or c 0 , c\ { , as specifications of the 
initial or final velocities of the molecules. Hence the problem of determining the 
final velocities of two molecules after an encounter, in terms of the initial velocities 
and whatever further independent variables are necessary to define the encounter, is 
equivalent to the determination of c' R in terms of c R and the variables of the 
encounter. Thus, in consequence of the invariability of c 0 , the velocity of the mass- 
centre, we need only consider the motion relative to G, i.e., the motion referred to 
uniformly moving axes with G as origin. 
* We here suppose that the effect of the external forces during the brief interval of encounter is 
negligible; this is legitimate if the gas is “ nearly perfect ” {cf. § 2). 
