436 Prof. Clausius on the Dynamical Theory of Gases. 
city, in other words, when wu is constant, and 5 alone variable 
from one molecule to another, the mean ae can be easily cal- 
culated. According to Prof. Maxwell, the value in question is 
r= Vur+02; 
whence, when u=v, follows r=v/2. This value is incorrect, 
however, as will be seen from the following considerations. 
Since all directions are equally probable for the molecules m, 
My, M5,..., the number of those whose lines of motion make 
angles between 3 and 3+ d3 with the line in which » moves will 
have to the whole number of molecules the same ratio that a 
spherical zone with the polar angle $ and the breadth dS has 
to the whole surface of the sphere, in other words, the ratio 
2a sin $d$: 4ar. 
The number of such molecules in the unit of volume is con- 
sequently 
N.4tsn3ds. 
Tn order to obtain the required mean value 7, the last expression 
must be multiplied by the relative velocity which corresponds to 
it, the product integrated between the limits o and 7, and the 
integral divided by N. Hence 
rai" Vu? +v?—2uv cos 3. sin S dS, 
0 
This gives at once 
1 3 
= [(u? + v? + Quv)® — (u? + v? — Quv)*], 
whence we may deduce 
2 
,u 
nT Umi when u <2, 
and 
pe 
ruts 7 when u> v. 
When u=z, both results coincide in value with 
r=, 
and thus verify my assertion. 
I remain, Gentlemen, 
Yours respectfully, 
Zurich, April 25, 1860, R. Ciaustius. 
