284 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
with respect to any orthogonal transformation of the co-ordinate axes. This places 
some restriction upon the possible modes of occurrence in F of (U, V, W) and of the 
space derivatives of T and (u 0 , v 0 , w 0 ), though not, of course, on the scalar quantities 
m and T. It is easy to see that the most general invariant function of the quantities 
involved in F must be compounded of the following elementary invariants :— 
( 11 ) 
C 2 = u 2 +v 2 +w 2 , 
(12), (13) 
(14) 
a _ du 0 dv 0 dw 0 
' “ dx dy dz ’ 
V 2 T = (— + — + 
“ v3« 2 dy 2 
DT = (U^+V^ + w|-)T, 
dx dy dz! 
(15) S' = U a ^+V ! ^+W=®y» 
ox dy dz 
vw{^ + |s')+wu(|fe+ 3m 
oy dz / \ dz 
dx 
dVn 
dm. 
°'+uv(p + bp,, 
ox dy) 
together with derivatives of the last four expressions formed by operating on them 
any number of times by the invariant differential operators V 2 and D, in the notation 
of (13) and (14). 
[, January 15, 1916. —Except in the case of highly rarefied gases, which were 
expressly excluded in § 2 (A), only the derivatives of the first order actually occur in 
F, to -the present degree of accuracy. The reasons for this will perhaps be more 
clearly apparent after reading §11, but the following considerations will elucidate the 
point. Whatever derivatives are contained in F must (§11) appear either in the 
equation of pressure or the equation of energy, so that, if the ordinary equations of 
viscosity and thermal conduction are to hold good, only the first-order space deriva¬ 
tives of temperature and mean velocity can be present; otherwise the ordinary 
coefficients of viscosity and conduction do not exist. In actual gases at normal 
densities the ordinary equations are shown by experiment to be valid; they fail, 
however, in highly rarefied gases because the terms in F which contain second-order 
differentials of T, u 0 , v 0 , w 0 are in this case comparable with those containing derivatives 
of the first order, as will be seen in detail in the future paper mentioned in § 2 (A). 
The coefficients of the first and second order derivatives respectively contain (\/l) and 
(A/0 2 , where A is the mean free path of a molecule and l is comparable with the scale 
of length within which the temperature and mean velocity vary appreciably ; except 
in rarefied gases {\/lf can be neglected in comparison with (A /l). The same inferences 
can be made also (cf. § 6) from the equations of transfer of § 3. 
For the present paper it is therefore sufficient (and it is convenient) to write down 
the following form of F forthwith :—- 
F = (u U + VU + W U) P, (C 2 ) + SP,(C 2 ) + S'P 3 (C 2 ), 
(16) 
