FREE AND PERFECTLY ELASTIC MOLECULES IN A STATE OE MOTION. 21 
immediately supplied by others of the same entering the stratum. The time taken 
by the set whose velocity is u and inclination to the plane 6 to traverse the breadth 
of the stratum is evidently as . , and in a unit of time the number of the impacts 
in the succession of these belonging to the set is proportional to u sin 6. But this 
is the value of the resolved velocity of the set. Referring back to the reasoning 
in §§ 2, 1G, the supporting effect of each impact on the heavy plane n was shown 
to be proportional also to the velocity of impact or molecular velocity resolved 
perpendicular to the plane. The supporting effect of each set in a unit of time is 
therefore as the square of the resolved or impinging velocity of the set. But the 
mean of all the square impinging velocities is jv z , and half the molecules in the 
stratum are continually approaching the plane ; the supporting effect of their con¬ 
tinuous action is therefore the same as would be derived from the medium reduced to 
half density advancing against the plane with the uniform velocity \/^v 2 . Now as A 3 
represents the number of molecules in the cubical unit of volume, the side of the cube 
being the unit of length, and \/ -\v l the number of such units traversed in a unit of 
time, the supporting effect of the medium on the heavy plane n in the unit of time, 
is the same as that derived from |>A 3 \/ j$v 2 molecules impinging with the velocity \f^v 2 . 
Hence it is obvious that A = -|A 3 \/ \v 2 , and io = \/\v 2 , or c = and 3 w 2 = v 2 . Also 
2 „ o v* A s o. 3 0 
n—~ iv 2 c A d = —— ; or v* A 6 = 3 an ; or v 
9 3 g ’ J 
= V 
3 9 
A 3 
Thus we obtain an expression for the square root of the mean square molecuk 
velocity in terms of the height of a uniform atmosphere — , or what is the same, in 
terms of the ratio of the number of molecules in a column of the medium of the height 
of a uniform atmosphere to the number in the column of a unit height: and since 
V2gh expresses the velocity acquired by a body in falling through the height h, we 
arrive at the following deduction. The mean square molecular velocity of a medium 
is equal to the square of the velocity that a body acquires in falling through one-ancl- 
a-half times the height of a uniform atmosphere; if the pressure of the medium is 
estimated from the effects of the molecular impacts on a perfectly rigid and elastic 
surface. If it were estimated by the effect on molecular elastic surfaces, there is 
reason to believe that the mean square velocity is double the amount specified 
(see Sec. 4).XI.* 
§ 18. Suppose that the cubical volume of the medium receives such an increment erf 
vis viva that under the constant pressure n the volume from 1 becomes 1 + — : it 
cpr 
has been shown in § 5 that the molecular vis viva must also from 1 have become 
* [The author here ai’rives at the correct conclusion, 
= .fh. —R.l 
