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APPENDIX I 



PRESSURE AND ENERGY 



IN the first part of this book it has several times been pointed out 

 that we can correctly calculate the pressure exerted by a gas if, in- 

 stead of ascribing to the molecules, as is actually the case, unequal 

 speeds that are constantly varying, we assume that they have all 

 the same mean speed. This mean value is of such size that the 

 kinetic energy on the assumption of equal speeds in the gas has 

 the same value as it really has with the actual inequalities that 

 exist. The justification of this simplifying assumption rests on 

 the fact that the pressure exerted by a gas is dependent on the . 

 speed of the molecules only in so far as it increases in proportion 

 to their kinetic energy. 



Against the validity of the reasons marshalled in Chapter II. 

 no objection can be raised. But it will not be superfluous to 

 calculate the pressure without this simplifying assumption. There 

 will therefore be made in the following investigation no assump- 

 tion of any kind with respect to the distribution of unequal speeds. 

 But the result of the calculation will be the same, viz. that the 

 pressure depends only on the mean kinetic energy. 



1*. Number of Molecules and their Paths 



Let the number of molecules of a gaseous medium in unit 

 volume be N. These N molecules do not all move with the same 

 speed, nor even in the same direction ; the components u, v, w of 

 the velocity of a molecule, reckoned along three fixed rectangular 

 axes, assume for different molecules values which vary from oo 

 to + oo. The number of molecules for which the values of the 

 components differ infinitely little from u, v, w, so that they lie 

 between the limits u and u + du, v and v + dv, w and w 4- dw, is 

 an infinitesimal of the order du dv dw ; it may be expressed by 



NF(u, v, w)du dv dw, 



A A 2 



