96 in. GENERAL PRINCIPLES. WORK AND ENERGY. 



The principle expressed by equations 60) will be referred to as 

 the Principle of Conservation of Moment of Momentum. On account 

 of the connection with the sectorial velocity it has received the 

 shorter and more euphonious title of the Principle of Areas. 



In case dU does not vanish, going back to equation 58), we may 

 divide out do and instead of 60) now obtain 



dH 



-jf = Z r (y r Z r -8 r Y r ), 



dH 



61) -jl = Z r (z r Xr-XrZr), 



dH 



where H x , H y , H z , have the same meaning as the left-hand members 

 of equation 60), but are not now constant. Stating in words: The 

 time derivative of the moment of momentum of any system with 

 respect to any point is equal to the resultant moment of all the 

 forces of the system about the same point. 



The equations 46) and 61) furnish us the six equations of 

 motion of a rigid body. Geometrically, we may say that the radius 

 vector of the hodograph ( 6) of the vector moment of momentum of a 

 system is parallel to the resultant moment of the forces acting on 

 tjie system at each instant of time, this statement being the com- 

 plement to the statement that the radius vector of the hodograph of 

 the velocity of the center of mass is parallel to the resultant of the 

 forces acting on the system. 



The three principles which we have now treated, the Principle 

 of Energy, the Principle of Motion of the Center of Mass, and the 

 Principle of Moment of Momentum, in the cases of conservation, give 

 us the first integrals of the equations of motion, and suffice for the 

 treatment of all mechanical problems. In the next chapter we shall 

 deal with a principle which is more general than any of these in 

 that it enables us to deduce the equations of motion and thus 

 embraces a statement of all the laws of Dynamics. 



