800 
PROFESSOR O. REYNOLDS ON CERTAIN DIMENSIONAL 
of the velocity may be different from the values of s for the density and mean square 
of the velocity, which latter are equal. 
Such a difference in the values of s, however, is not important as regards the 
immediate results of this investigation, and in the absence of any evidence to the con¬ 
trary all values of s will be treated as equal. 
The mean component values of s and general expression of cr(Q).— Gas continuous. 
82. When the gas is continuous, by theorem (II.) all values of <r(Q) for the groups 
A, B, C, &c., at any point may severally be expressed as functions of p, a, U, V, W for 
this point, their space variations, and 5. 
The first step is to express as a function of p, a, U, Y, W, the elementary portion of 
cr(Q) belonging to an elementary group of molecules, and then to integrate for the 
largeT groups. 
Putting cr'(Q) for the value of o-(Q), which would result from the resultant uniform 
gas at a point, and Scr(Q) for the elementary portion of cr(Q) belonging to an 
elementary group whose directions are l, m, n, .then since p, a, U, Y, W, for any 
point x, y, z, are functions of x, y, z , Scr(Q) is a function of x, y, z, and for the point 
x-\-ls, y-\- ms, z-\- ns we have 
So-(Q)=S 0 -'(Q)+s(«£+m| y +»|)So-'(Q).(38) 
together with terms which are neglected as small. 
Whence integrating for all the groups in A, and putting A for o-'(Q) 
cr"(Q) = A+s( [*{l~ -\~ni~ -\-n^r ^SA sin 0d9d(b ..... (39) 
J o J o \ dx dy dzj 
where cos 6=1, m= sin 6 cos <£, n— sin 6 sin (f>, and similarly for the other seven 
groups, B, C, &c. 
The values of A, &c., are given in Table XX. 
The integrals of ,S'| (ly -j-' m f + 4 ) sa } sin 6d9d(j) will involve terms which will 
be the differentials of the corresponding terms in Table XX., multiplied by s and by 
certain numerical coefficients which are the mean values of l, of Id divided by l, and 
so on, and which may be written 1, y, yy, &c. The values of s multiplied by these co¬ 
efficients are the mean component values of s for p, a, a 3 , &c., for the groups A, B, C, &c. 
As it is these component values which come into comparison with the distances from a 
solid surface, it is important to preserve the coefficients, therefore instead of using the 
