160-164] GENERAL PRINCIPLES. 147 



162. Conservative Field. As in Article 80 let m be the 



mass of the particle, x, y, z its coordinates at time t, X, Y, Z the 

 resolved parts of the forces acting upon it ; then, if the field of 

 force is conservative, Xdx + Ydy + Zdz is the complete differential 

 of a function W, so that 



" > * > " 



dx dy dz 



The equations of motion of the particle are 



mx = , my = -^ , mz = -^ , 

 dx dy dz 



and they possess the integral (Article 151) 



\m (a? +f + z 2 )=W+ const. 



If v is the magnitude of the velocity of the particle at time t, 

 this equation can be written 



1 TO? ;2 = jp + const. 



This equation applies to all the particular cases discussed in 

 Chapter IV 7 . 



163. Conservation of Linear Momentum. Suppose the 

 axis x is a direction in which there is no resolved part of the 



brce acting on the particle. The equation of motion by resolution 

 )arallel to the axis x is mx = 0, and it follows that x is constant 

 hroughout the motion, or the resolved velocity in any direction in 

 wrhich the resolved part of the force vanishes is constant. This is 

 special case of the general principle considered in Article 111. 



164. Conservation of Angular Momentum. Suppose 

 he axis z is a line about which the forces acting on the particle 

 lave no moment. Then we have 



xY-yX = Q. 



Hence multiplying the equations 



mx = X, my = Y 

 espectively by y and x, and subtracting, we have 



m (xy - yx) = 0, 

 nd this equation possesses the integral 



xy yx = const. 



This is a special case of the general principle considered in 

 Article 112. 



102 



