TRANSACTIONS OF SECTION A. 717 
the volume common to the two spheres is = 7R®—7ROs, or = 7R (1 - a 
So the value of € for the new sphere is g- + = wv, where x is a vector for 
which positive and negative values are equally probable, and for which on 
ye d 3 3 : : 
average 22=£. So we find = (vx—&)= ~ gp OD average if € be given, 
2 EX 
In the same way we find the mean value of ) to be = that is, proportional 
to &, 
3. Now consider the function 
h 9 ° ¥o dé 
M= iit andyde| [fe Kong 05? + 183") dw,dwydwww, © j 
C7 
in which w, w, w are the component velocities of a molecule, and the iategration 
includes all space and ali values of w, w, w.. If we follow individual molecules, 
W, Wy w: Yemain unchanged in the absence of collisions. But if we regard w, wy wz 
as belonging to those molecules which are for the time being within a fixed space, 
W, W, w vary with the time by the passage of molecules into or out of the space. 
dM 
Now in stationary:motion M is constant, and at 0, that is 
{V Boel edw, dw dw x { swat, = + 2 = (yz) =0, in which 
W= U2 +0, +0, , : : . : : ; 5 c : ce a ley) 
‘We have now to express . ; (w,,W:). 
Suppose, near a certain point P, ES 
~ 
in 
is positive. 
Form the integral I for a small cylindrical space AB, containing P, whose ends 
are unit area of two planes A and B, respectively parallel to ay. Then we find 
dé d Qhwt 3 & 
that — on (w,w-) has throughout AB the mean value — —— - is 
iz 
Bye) 
And therefore 
2 0 |{faxayae| | soe A(w," + wy? +w “dwydw, dre, 
i dd& 2w*t 3 & 
eW,— —— ——— _-« — e . . Il, 
Ciera Sao Te un) 
Transition to Finite Spheres. 
4, We now pass to the case in which our molecules, instead of being material 
points, are finite spheres, each of unit mass and diameterc. The first effect of 
this alteration is to increase the quantity of momentum transferred across any 
plane per unit of area and time in the ratio 1: 1+, where k=2zc*p. But this 
increases both the terms of I or II in the same ratio, and therefore ae so far as 
this is concerned, remains zero. 
___ 6, But collisions alter the term 7 (wa), because at each collision the direc- 
tion of motion changes for each of the colliding spheres. The state of the medium 
near P being as assumed in Art. 3, viz. & positive, consider two planes, one the 
