8 
MR. S. H BURBURY ON THE APPLICATION 
And now writing T, for T, and T, for the energy of the stream motion whose vertical 
component is U, we find that T + kT,. is constant throughout the column, and we 
may now write T + *T r = 3/2 h. 
12 . It follows that we cannot express the law of distribution of velocities among 
+ v 2 + w 2 
the spheres in the form Ce -7,T , or Ce -A5 2 , as in the rare medium, with 
T = 3/2 h. Tlie law must be Ce _/iQ , in which Q is some quadratic function of the 
velocities. Suppose that for n spheres it is 
Q„ = a x u 2 + ^\2 u i u 2 d~ c h u z + & c -> 
with corresponding expressions for the components v and w. 
If we find the mean value of Q„ by integrating e~ hQln for all values of iq, u 2 , &c., 
between the limits i 00 , we find, there being 3 n variables or 3 component velocities 
for each sphere, 
Q„ — Bn/2h. 
But 
3n/2h = 11 (T -4- ^T,.), 
therefore, 
Q« = n (1 + /cT,.). 
This result might be considered to justify the assumption that the law of distri¬ 
bution of velocities is in all cases, including the field of no forces formed by making 
f — 0, accurately expressed as follows. The chance that the velocities of n spheres, 
forming a group together, shall be u v . . . iq + du h &c., is proportional to e~ , ‘ il(T + * Tr) , in 
which 
nT = 4 MS (u 2 -f u 2 + w 2 ) 
nT,- = % MS {(u — u') 2 + (v — r')-' -f (iv — iv') 2 }/n. 
But we must remember that the whole treatment is based on the consideration of 
a great number of spheres, so that we cannot safely assume the law to hold when n 
is small. Let us, then, consider the subject from yet another point of view. 
13. I have shown elsewhere (‘ Science Progress,’ November, 1894), that in a dense 
medium the velocities of contiguous spheres cannot be independent of one another, 
because there is a presumption that recent collisions of two spheres near to one 
another have been with the same third sphere, and they have, so to speak, inherited 
some common velocity from it. In other words contiguous spheres have been exposed 
to the same environment, and, in the dense medium, environment does not change 
rapidly ; therefore their velocities are not independent. 
If, therefore, the spneres contained in a volume V be n in number, and their 
positions known, the chance that their velocities shall be 
