with Remarks upon the Mechanical Theory of Heat. 89 



portion to the radius of the sphere of action as the entire space 

 occripied by the gas, to that portion of the space which is actuaUy 

 filled up by the spheres of action of the molecules. 



(7.) la order to have a tlefiuite numerical example^ let us 

 assume, in round numbers, that the spheres of action of the 

 molecules are so small that only a thousandth of the space occu- 

 pied by the gas is really filled out by the spheres of action, and 

 that the whole remaining space be free for motion. 



For this case we have 



-T^o=1000, 

 whence it follows that 



-=16a3 (8) 



On applying these values we obtain from equations (6) and (7), 



/' = 1333p = 83X, (9) 



/=:1000p = 62\ (10) 



The first expressions in both equations show that, under the 

 assumption made, the mean path has a considerable length in 

 comparison to the radius of the spheres of action, and that there- 

 fore, as far as the effect of this circumstance is concerned, Ma- 

 riotte and Gay-Lussac's law may be very nearly true for the gas. 

 By a simple calculation it may be shown that the relation of 

 1000 to 1 completely suffices, even for those approximations 

 found by Regnault with permanent gases. It follows that the 

 magnitude of the spheres of action which was taken for illustra- 

 tion, although arbitrarily chosen, may yet be regarded as one 

 within the bounds of possibility. 



But if we now regard this same mean value of the length of 

 path in such a manner as to compare it, not with the sizes of 

 molecules, but with our usual units of length, we obtain totally 

 different relations. In all physical and chemical investigations 

 in which o])portunity presents itself for drawing conclusions 

 concerning the weight and size of the separate molecules, we are 

 invariably led to the conclusion that, compared with all measur- 

 able magnitudes, molecules must be of extraordinarily small 

 size. yVs yet, no one has been able to establish a bounding line 

 on the other side (for smallness). Accordingly, when an ordi- 

 nai-y unit of measure, e. g. a litre, is filled with gas at the ordi- 

 nary atmospheric pressure, we must assume that the number of 

 molecules present is very great, and that consequently the di- 

 stances between the molecules is very small. Accordingly the 

 values previously found for /' and /, namely, 83X and 02X, must 

 only be regarded as small magnitudes. 



