382.] FREE SYSTEM. 



213 



380. Exercises. Show the existence of a force-function and find its 

 expression in the following cases (comp. Art. 86) : 



(1) When the resulting force F at each particle m is constant in 

 magnitude and direction (gravity). 



(2) When the forces F are all attractions, each being directed to 

 some fixed centre O and a function of the distance r from this centre. 



(3) When the forces F are the mutual attractions of the particles 

 constituting the system. 



381. A variable system of n particles possesses a centroid 

 whose co-ordinates x, y, ~z satisfy the equations 



M'X ^mx, M - y = ^my, M -~z = lanz. 



The developments of Arts. 226, 227, in particular the principle 

 of the conservation of linear momentum, or the principle of the 

 conservation of the motion of the centroid, hold for a variable 

 system just as well as for a rigid body. The position of the 

 centroid in the system is of course variable with the time. 



The principle just referred to asserts that, if 2X = o, 2Y = o, 

 3Z = o, the centroid of the system is at rest, or moves with con- 

 stant velocity in a straight line. It should be noticed that the 

 conditions 2^=o, 2F=o, ^Z=o do not mean that there are 

 no forces at all acting on the system ; they only mean that the 

 resultant of these forces reduces to zero while there may be a 

 resulting couple different from zero. The principle would, for 

 instance, apply to the solar system if the action of the fixed 

 stars be regarded as vanishing or as reducing to a couple ; the 

 mutual attractions of the various members of the system occur 

 in pairs of equal and opposite forces, and have, therefore, a 

 resultant zero. 



382. Similarly, the developments of Arts. 228-232, in par- 

 ticular the principle of the conservation of angular momentum, 

 or of areas, and the properties of the invariable plane, apply 

 without change to the free variable system. 



