106 GENERAL THEOREMS. [CHAP. VI. 



105. Equations of motion of a system of particles. 



Let Wi be the mass of one particle of the system, sc lt y lt z l its 

 coordinates at time t, X lf Y lt Z 1 the sums of the resolved parts 

 parallel to the axes of the forces exerted on this particle by 

 particles not forming part of the system, X/, F/, Z{ the sums of 

 the resolved parts parallel to the axes of the forces exerted on the 

 same particle by the remaining particles of the system. 



The equations of motion of this particle are 



m& = X, + Xj, mSi =Y,+ F/, m& = Z, + Z{. 

 Similarly the equations of motion of a second particle of mass 

 w 2 at (# 2 , ya, ^2) may be written 



W2# 2 = X 2 + X/, mjj z = F 2 4- F 2 ', m^ = Z 2 + Z. 

 We shall write as the type of such equations 



Then (X, Y, Z) is the type of the external forces, and 

 (X' } Y', Z') is the type of the internal forces. 



106. Law of Internal Action. The system of the internal 

 forces between the parts of a system is equivalent to zero. 



Since the action between a pair of particles of a system is a 

 pair of equal and opposite forces in the line joining the particles, 

 the resultant of all these forces regarded as vectors localised in 

 these lines is zero, the sum of their resolved parts parallel to any 

 line is zero, and the sum of their moments about any axis is zero. 



Thus, in the notation of the last Article, one of the forces con- 

 tributing to the resultant whose resolved parts are X/, F/, ZJ is 

 the action of m 2 on m l5 and one of the forces contributing to the 

 resultant whose resolved parts are X 2 f , F 2 ', Z. 2 f is the action of m l 

 on ra 2 . These are equal and opposite forces in the line joining m l 

 and ma. The resultants whose resolved parts are the forces 

 typified by X', Y', Z' are made up entirely of such components. 



Hence we have 2Z' = 0, 2F' = 0, 2' = 0, and 

 2 (yZ' - *F') = 0, 2 (zX' - xZ') = 0, 2 (xY' - yX') = 0. 



107. Simplified forms of the equations of motion. 



Adding the left-hand members of all the ^-equations of motion, 

 and remembering that 2Z' = 0, we obtain Sraa; = 2 



