604 Mr. J. H. Jeans on the 



With our present conventional basis of probability, we have 

 only to ask " For what proportion of the systems represented 

 in onr generalized space is the condition satisfied that there 

 shall be a molecule such that x lies between x and 

 x-\-dx. . . &c., u lies between u and u + du &c? " Now for 

 a certain number of systems this condition will be satisfied 

 by molecule A. These systems will be those for which 

 x a lies between x and x + dx ; and similar conditions are 

 satisfied by y a , z a , u a , v a , w a . These systems together form an 

 element of the generalized space 



dx dy dz du dv div \\\ . . . dx h dx c . . . dy h dy c , . . (4) 



the integration being with respect to all the 6N variables 

 except x a y a z a u n v a iv a , and extending over all values except 

 such as are excluded by equation (3). Those systems for 

 which the conditions are satisfied by molecule B form an 

 equal and similar element, and so also for every other of the 

 N molecules. The probability which we require is the ratio 

 of the sum of these N elements to the whole volume of the 

 generalized space, and is therefore 



^dxdydzdudvdiv J^'^^l^: . . . (5) 

 jjj . . . dx a dx b dx c . . . 



§ 11. This is the probability when nothing at all is known 

 about the gas. Let us now suppose that the velocities of the 

 individual molecules are known, and calculate the probability 

 in this case. An element such as (4) will not now be con- 

 tributed by every molecule : there will be an element contri- 

 buted by each molecule of which the velocities lie within the 

 assigned limits, and the number of these may be taken to be 

 N/'(i«, v, w) du dv dw. The element of volume is not given 

 by expression (4), but by 



dx dy dz\$ . . . dx b dx c . . . dy b dy c , ... (6) 



the velocity-coordinates not entering into the integration 

 at all. We now find instead of expression (5) a probability 



N/(«, v, w) dx dy dz du dv dw^ M''' dXbdd ' c , . (7) 



^...dx a dx h dx c ... 



§ 12. In the special case of a gas in which the radii of the 

 molecules are very small, the regions which are excluded by 



