136 HEAT. 



be Zmuj^ x -^ = mu^. If we have in all n molecules, their velocities 



a 



perpendicular to A being u v u 2 . . . u n and if the total momentum 

 imparted per second to A is equal to p, 



Considering the face B we similarly obtain 



p = m(v* + v 2 z + 

 and on 



p = m(w 1 2 + w.?+ 

 Adding these three together, 



. . . . + m(u n * + w n 2 



If V 2 is the mean of the squares of the velocities, then we may write 

 this in the form 



3p = mnV 2 

 But mn is the total mass of gas in 1 cc., that is, is equal to the density p. 



Hence 3 = />V 2 or V 2 = 5? 



P 



V is not the mean velocity, but the square root of the mean of the 

 squares of the velocities. It is termed " the velocity of mean square," 

 and it may be shown that it is somewhat greater than the mean 

 velocity. 



Maxwell investigated the law of distribution of velocities about the 

 mean which would justify the supposition of constancy of distribution, 

 and he showed that it is of " exactly the same mathematical form as the 

 distribution of observations according to the magnitude of their errors, 

 as described in the theory of .errors of observation. The distribution of 

 bullet-holes in a target according to their distances from the point aimed 

 at is found to be of the same form, provided a great many shots are fired 

 by persons of the same degree of skill" (Maxwell's Theory of Heat, 

 p. 309, ed. 5). It can be shown that the mean velocity U is given by 



or very nearly U = ^ V 



19 



If the gas is hydrogen at 0, then if p 1014000 (the value of 76 cm. 

 of mercury in dynes per sq. cm.) p is very nearly -00009. 



Whence V= 184000 cm./sec. nearly; 



and U= 170000 cm./sec. nearly. 



If we take any other gas at the same temperature and pressure the 



velocity is inversely as the square root of the density. Hence for oxygen 



we must divide by 4, and 



V = 46000 nearly, 

 U = 42500 nearly. 



