114 Prof. R. Clausius on the Theorem of the Mean Ergal, 



the time, but during the small portions required for the collisions 

 is variable, we take its mean value, which, as before, we will 

 designate by w. This w has different values for different points ; 

 and hence, if ft? denotes the arithmetic mean of all occurring 

 values of w, we can put 



ur=Wz% (82) 



wherein z is a quantity which, for the different points, may vary 

 between and oo . We will refer the law to this quantity, 

 representing the probability that for an arbitrarily selected point 

 it lies between the value z and the infinitesimally different value 

 z-\-dz by/(2"), understanding by /a given function. 



If, in correspondence with the assumptions made in the pre- 

 ceding section, the number of points present be denoted by ^N, 

 the law can also be expressed thus : — The number of the points 

 for which that quantity lies between z and z + dz is equal to 



If the expression be integrated from z = to z=oc , the total 

 number J N must be obtained, whence it follows that the func- 

 tion / must satisfy the equation 



I 



/(*)&=! (83) 



If for any quantity (preliminarily denoted generally by q) de- 

 pendent on w, and which can therefore, when W is given, be 

 considered as a function of z, we wish to determine the arithmetic 

 mean of all the values occurring, we have to multiply \Nf{z)dz 

 by q, integrate the expression from to oo , and divide the inte- 

 gral by JN. The mean value of q has consequently for its ex- 

 pression, since ^N is eliminated: — 



J" 



gf{z)dz. 



Applying this specially to the quantity w itself, for which Wz 2 

 can be written, and recollecting that the mean value thus deter- 

 mined must be equal to tt>, we get a second condition-equation 

 which the function / must satisfy — namely, 



£ 



zj(z)dz=l (84) 



Now, with respect to the form of the function /, Maxwell, as 

 is well known, in his excellent "Illustrations of the Dynamical 

 Theory of Gases" *", deduced from the rules of probability a most 

 important law for the velocities of the molecules. According to 

 it, for an arbitrarily selected molecule, the probability that the 



* Phil. Mag. S. 4. vol. xix. p. 19. 



