PREFACE. 



THE usual point of view in the study of mechanics is that 

 where the attention is mainly directed to the changes which 

 take place in the course of time in a given system. The prin- 

 cipal problem is the determination of the condition of the 

 system with respect to. configuration and velocities at any 

 required time, when its condition in these respects has been 

 given for some one time, and the fundamental equations are 

 those which express the changes continually taking place in 

 the system. Inquiries of this kind are often simplified by 

 taking into consideration conditions of the system other than 

 those through which it actually passes or is supposed to pass, 

 but our attention is not usually carried beyond conditions 

 differing infinitesimally from those which are regarded as 

 actual. 



For some purposes, however, it is desirable to take a broader 

 view of the subject. We may imagine a great number of 

 systems of the same nature, but differing in the configura- 

 tions and velocities which they have at a given instant, and 

 differing not merely infinitesimally, but it may be so as to 

 embrace every conceivable combination of configuration and 

 velocities. And here we may set the problem, not to follow 

 a particular system through its succession of configurations, 

 but to determine how the whole number of systems will be 

 distributed among the various conceivable configurations and 

 velocities at any required time, when the distribution has 

 been given for some one time. The fundamental equation 

 for this inquiry is that which gives the rate of change of the 

 number of systems which fall within any infinitesimal limits 

 of configuration and velocity. 



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