Illustration of the Isothermal Formula, 303 



sider a single molecule a; let there be in volume V a number 

 NF (v, w) dw of molecules which are moving with relative 

 velocity > w and < tp + <i«? with respect to molecule a. Call 

 -v/r the acute angle between the iv and R directions at the 

 beginning moment ol encounter. It is then easily proved 

 that the molecule a will suffer per unit time 



9_T\JT)2 



— ^ — ivF(v, w) sin yjr cos ty dw dyjr . . (1) 



encounters of the w, yfr description. Kow in a large class of 

 laws of force the duration-time of entanglement is determined 

 by the values the two variables w, yjr assume at the beginning 

 moment of encounter ; and from the following it will be 

 obvious that our deduction would apply with but formal 

 alterations to the more general case, when further variables 

 are required. Let then t be the duration-time of entangle- 

 ment ; every encounter of the above class (1) will belong to 

 the class of T-encounters. These encounters are happening 

 with the concurrence of molecule a ; but the same being true 

 for every other member of the v-class, and the state of things 

 in the whole mass being assumed to be steady, we shall find 

 the number of encoun fcers which at any time are just happen- 

 ing in the volume considered by multiplying (1) by t and 

 by ~Nf(v)dv, by integrating, and halving. Let \~Ec repre- 

 sent that number ; c therefore is taken to mean the ratio of 

 molecules which belong at a given time to molecular systems. 

 The number of free molecules is equal to N(l — c) and 



c = 



27rNR 2 

 V 



J j j TW f( v ) -^ ( v i w ) sm ^ cos ^ dv dw d^r. . (2) 



2. Let us now proceed to the application of the virial 

 equation. In this equation we have to introduce, it is well 

 known, certain average values of the kinetic energy and the 

 virial, and these averages are to be found in the following- 

 way. We have to write down the value_ of the kinetic energy 

 and the virial for every molecule, to sum them up, to continue 

 this process during some long period of time, and to average 

 at last all the instantaneous sums we have computed. ^Now, 

 since the state of the gas is steady, we are justified in apply- 

 ing the theorem directly to the instantaneous sums of kinetic 

 energy and virial, which cannot differ materially from their 

 time-averages. There is no difficulty in performing the sum- 

 mation as far as the kinetic energy of free molecules and the 

 virial of external pressure (no other external forces being- 

 admitted) are concerned ; but terms relating to molecular 



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