86 



The supposition would now natumlly suggest itself, 1"'. that the 

 mean value of the kinetic energy for this degree of freedom would 

 be the same for the ditferent points of the region v; 2'"'. that the 

 partition of the velocities for this component would be represented 



by Ce (Ir ^). Tiiis expression however leads to an untenable 



formula for the distribution in configuration. I will therefore assume 

 that the ladial component of the velocity does not follow Maxwell's 

 law for the distribution of velocities, and that the mean kinetic 

 energj^ belonging to it is different for different points of the region z;. 



For points at a distance r from the centi-e e.g. it will amount to 

 F {)'). This function F (r) is unknown; we oidy know that its mean 

 value for different values of r will amount to 5 U. The components 

 of the velocity perpendicular to the radius vector will be denoted 

 by .s'' and i . Their mean kinetic energy will be equal to the normal 

 equi partition-amount. 



In a harmonic vibration kinetic and potential energy are jieriodi- 

 cally converted into one another; Ihei-efore the distribution in 

 configuration will follow the same law as the distribution in velocity. 

 We are therefore justified in the following assertions concerning it. 

 Let us take all molecules with a definite velocity y, and investigate 

 their deviations from the position of equilibrium. We will call the 

 component of the deviation in the direction i> >v, the components 

 nornuil to this direction r^ and r, . The mean value of {fr/ for 

 these molecules \yill again amount to F[\>), which function agaiji is 

 unknown, whereas its average value for ditferent values of v» amounts 

 to \ U. The average values of hf^'s^ and kp'i' pj'esent the normal 

 equipartition-amount. 



In this way we are induced to represent the number of particles 

 whose coordinates and momenta are included between definite 

 limits by : 



fy.+ f 



Ne ■/ {mr, ms', iiit', 7\; r^, r^, r) </iin' dins' dint' dr„ d g dvi . . . (14) 



Here Ej, ■= 72 "^ ('"^ 4" ^'^ ~\~ ^'0 ^^^^ ^<j =^ the potential energy. In 

 the space V this potential energy has a constant amount e^ ; in the 



i) These Proc. Vol. XV, p. 1355 I really have expressed the opinion that this 

 partition of velocities would probably exist. 1 have however written erroneously 

 '/s U in the numerator of the exponent instead oi U. ]i U represented the total 

 kinetic energy of a particle with three degrees of freedom, 1/3 U would be the 

 right value. U re^jresents however the kinetic energy for one degree of freedom. 



