730 
PROF. 0. REYNOLDS ON THE FORCES CAUSED BY THE 
These results, although they do not pretend to he more than approximate, show how 
exceedingly small is the real effect, and they place these phenomena of motion caused 
by heat in a light from which the exceeding delicacy and sensitiveness of the instruments 
have altogether withdrawn them. 
The difference of Temperature. 
Having obtained these measurements of the force, it remained to see what difference 
of temperature would be necessary, according to the kinetic theory, that the reaction 
from the communication of heat might equal these forces, and then to ascertain how far 
such a difference of temperature actually existed. To do this I have had to enter upon 
new and somewhat doubtful ground : however, I venture to submit the following, which, 
although it contains assumptions, contains none but what are legitimate and strictly in 
accordance with the kinetic theory. 
Theoretical Difference of Temperature. 
Whatever may be the nature of the action by which heat is communicated from a 
surface to a gas, the result, according to the kinetic theory, is to increase the mean square 
of the velocity with which the molecules move hi the ratio of the temperature : thus, 
if v be the initial velocity, and r the initial absolute temperature, and if 
v 2 =A r, 
where A is a constant depending on the nature of the gas, then 
(' v-\-dv) 2 =A{r-\-dr ), 
or, neglecting dv 2 as a small quantity, 
Adr=2vdv, 
dr=2dv T - 
v 
Now, if we assume that each molecule comes up to the surface with a velocity v, and 
leaves with a velocity v-\-dv, we shall have the greatest reactionary force which it is 
possible that the heat could produce. That the force produced is as large as this is not 
probable. We know that at ordinary densities the molecules communicate the heat to 
each other, so that they do not come up to the surface with so small a velocity as v. The 
smaller the tension of the air, however, the less will be the difference ; so that the force 
which we have < assumed is the limit towards which its force tends as the vacuum 
improves, so long as the conditions of a perfect gas are fulfilled*. The increase dv in 
the velocity with which the molecules leave the surface would increase the pressure in 
the ratio 
2v + d v 
or p + dp dv 
p ‘ 2v 
* Proc. Roy. Soc. 1874, vol. xxii. p. 407. 
