90 in. GENERAL PRINCIPLES. WORK AND ENERGY. 



On the other hand 



40) = _ 1? _ IE _ du 



Consequently 



Thus 39) becomes 



41) lmr - + i> m r -f + vm r - = 0. 



But since A, p, v are perfectly arbitrary this is equivalent to the 

 three equations 



Since the w's are independent of the time, we may differentiate 

 outside of the summation and write the above 



d* d* d 2 



43) -^ 2 r m r x r = 0, j^ Z r m r y r = 0, -^ 2 r m r g r = 0. 



If we define the coordinates of a point x, y ~, ~z by the equations 



and if we consider a mass m to consist of m particles of unit mass, 



being the sum of the x- coordinates of the whole number of unit 

 particles divided by their number is the arithmetical mean of the 

 x- coordinates. If m is not an integer, by the method of limits we 

 extend the ifiotion of the mean in the usual manner. The point 

 x, if, ~s, the mean mass point thus defined is called the center of mass 

 of the system. (The common term center of gravity is poorly adapted 

 to express the idea here involved and had better be avoided. We 

 shall see in the chapter on Newtonian Attractions that bodies in 

 general do not possess centers of gravity.) 

 The equations 43) thus become 



A A\ d*x ^ d*y d* z ^ 



44 ) 5F=> d^ = > 5F- a 



Therefore the center of mass of a system whose parts exert forces 

 upon each other depending only on their mutual distances moves 

 with constant velocity in a straight line. This is the Principle of 

 Conservation of Motion of the Center of Mass. It evidently applies 

 to the solar system. What the absolute velocity of the center of 

 mass of the solar system is or what its velocity with respect to the 

 so-called fixed stars we do not at present know. 



