statistical iiiechanics 



35 



shortly within the doniinion dil in space and liaving velocity components within 

 the limits 



«1 ± -J d'^i 

 ± dh\ 



w-^ ± ^ dw-^ 



shortly within rfs^. 



The frequency function is a function of t and moreover of x, «/, 2\ m^, x\, w^. 

 Consider the integral 



■ (54) F = jQf,d,,dil 



integrated over all values of the velocities all over the space Ü where stars are to 

 be found. 



The limits of ß change with the time partly because the assembly of stars is 

 moving in space, partly because the volume is continuously changed with the time. 

 The limits of the integral are hence dependent on the time. 



If N is the whole number of stars within ß, so is 



(55) N^jf.dB^dÜ 



where also the integration is to be performed over all values of the velocities and 

 the coordinates. 



We hence have 



(56) F=NM(Q), 



where M{Q] denotes the mean value of Q. 



It may be observed that F depends on no other variable than the time and 

 that iV is a constant. 



We propose to form the differentialcoefficient of F, in regard to t. 



The time enters into the right member of (54) in the functions Q and 

 under the integral, and moreover in the integral, which is to be taken over the 

 whole dominion We thus get 



(57) 



dF 

 ~dt 



9. 



dt 



ds^ d9. 



Ài 



dû. 



where the latter integral (multiplied by dt) 

 denotes the change in F caused by the 

 change in the limits of 9.. If in fig. 5 the 

 dotted figure denotes the limits (in 3 dimen- 

 sions) after the time dt, so this integral is 

 obtained by integrating Qf-^ over the dominion denoted by + and subtracting the 

 integral over the dominion denoted by — . 



Fig. 5. 



