10 CONSERVATION OF 



efficient to indicate that they are to be regarded as constant 

 in the differentiation. 



We may give to this principle a slightly different expres- 

 sion. Let us call the value of the integral 



JT. 



.dp n d qi ... dq n (23) 



taken within any limits the extension-in-phase within those 

 limits. 



When the phases bounding an extension-in-phase vary in 

 the course of time according to the dynamical laws of a system 

 subject to forces which are functions of the coordinates either 

 alone or with the time, the value of the extension-in-phase thus 

 bounded remains constant. In this form the principle may be 

 called the principle of conservation of extension-in-phase. In 

 some respects this may be regarded as the most simple state- 

 ment of the principle, since it contains no explicit reference 

 to an ensemble of systems. 



Since any extension-in-phase may be divided into infinitesi- 

 mal .portions, it is only necessary to prove the principle for 

 an infinitely small extension. The number of systems of an 

 ensemble which fall within the extension will be represented 

 by the integral 



/ . . . / D dp! . . . dp 



If the extension is infinitely small, we may regard D as con- 

 stant in the extension and write 



D I . . . I dp l . . . dp n dq^ . . . dq n 



for the number of systems. The value of this expression must 

 be constant in time, since no systems are supposed to be 

 created or destroyed, and none can pass the limits, because 

 the motion of the limits is identical with that of the systems. 

 But we have seen that D is constant in time, and therefore 

 the integral 



I . . . / fa . . . dp n dq l . . . dq n , 



