48 AVERAGE VALUES IN A CANONICAL 



which is the value of the same integral for infinite limits. 

 Thus the probability that the value of x^ lies between any 

 given limits is expressed by 



C 

 J 



e 2& d Xl . (125) 



The expression becomes more simple when the velocity is 

 expressed with reference to the energy involved. If we set 



s=(^x l , 



the probability that s lies between any given limits is 

 expressed by 



~ S *ds. (126) 



Here s is the ratio of the component velocity to that which 

 would give the energy ; in other words, s 2 is the quotient 

 of the energy due to the component velocity divided by . 

 The distribution with respect to the partial energies due to 

 the component velocities is therefore the same for all the com- 

 ponent velocities. 



The probability that the configuration lies within any given 

 limits is expressed by the value of 



M f (27r) f . . . /**.** . . . dz v (127) 



for those limits, where M denotes the product of all the 

 masses. This is derived from (121) by substitution of the 

 values of the integrals relating to velocities taken for infinite 

 limits. 



Very similar results may be obtained in the general case of 

 a conservative system of n degrees of freedom. Since e p is a 

 homogeneous quadratic function of the ^>'s, it may be divided 

 into parts by the formula 



_ 1 ^^p -I @p /-I OQ\ 



