40 Motions of a System of Bodies. 



Art. IV. — Motions of a System of Bodies; 



by Prof. Theodore Strong. 



Continued from p. 345, Vol. xxn. 



. * 



. Let m, m' 9 m". &c. denote the quantities of matter in the moving 

 bodies, or the number of times which they severally contain the as- 

 sumed unit of masses : supposing them so small that every unit of 

 each may be considered as acted on by the forces (which are sup- 

 posed to affect their motions,) with the same intensity. 



1. Motions of the system when estimated in the directions of three 

 fixed rectangular axes drawn (at pleasure,) through any assumed 

 point for their origin. 



Let the coordinates be designated by the axes of a?, y, z, sev- 



* 



erally : put a?, y, *, a/, y 7 , #', &c. for the coordinates (reckoned 



from their origin,) which define the places of m, m', &e. at any time t. 



Let P, Q, R, P', Q', R', &c. be the resultants of all the forces 



which affect a unit of w, mf 9 &c. when reduced to the directions of 



the axes of oc, y, z; severally ; then by known formulae, supposing 



d 2 x d 2 x* 



dt=const. ((a) p. 284, vol. xvi.) ^i=P 3 ^r=P', &c. (1); 



d 2 y d 2 y' d 2 z d 2 z' 



5^ =Q > W^W* &c> ' W* ^ =R ' !F =R '> &c -( 3 )5 wllich 



, are the equations of motion required. 



Q 



&c. are the resultants of all the accelerations or retardations which 

 m, m' 9 -$zc. receive, whether from their mutual attractions or repul- 

 sions, or from bodies foreign to the system, also that the reactions 

 of the surfaces or curves on which any of the bodies may be suppo- 

 sed to move are included ; it is also supposed that P, Q, R, &c. tend 

 to increase x 9 y, z, &c. ; but should any of them tend the contrary 

 way their signs must be changed. The forces which arise from the 

 actions of the bodies on each other may be made to # destroy each 

 other by the following method. 



Let p denote any force which a unit of m exerts on a unit of m\ 

 then it is evident that a unit of mf will react on a unit of m with the 



force 



known 



law of action and reaction ; hence m p=the whole force of m on a 

 unit of m', and - mf p is the whole force of the consequent reaction of mf 

 on a unit ofm; if m p' equals the value of m p when reduced to the 

 axis of *, then evidently — m f p' equals the value of— m / p when re- 



