13] FORCE. STATIC AND. KINETIC REACTIONS. 25 



the two particles , fixing our attention on the facts embodied in 

 equations 40). Had we been merely astronomers this is what we 

 should have been obliged to do. We may perhaps doubt whether 

 we should have in this way arrived at the conceptions of force 

 which we possess with the aid of both senses. At any rate no one 

 can doubt that an individual newly arrived in this world learns its 

 properties as much through the muscular sense as through the more 

 generally appreciated sense of sight. 



Let us now reverse the mode of looking at equations 41). 

 Suppose that we find that under given conditions a certain agency 

 will produce a certain force, as shown by the motion of some body, 

 and suppose that as the circumstances are changed we can always 

 measure the force. If then it is possible to submit a second body 

 to the action of the same agent under similarly varying circumstances, 

 we shall be able to find the motion of the second body. The 

 equations 41) under these circumstances furnish merely another 

 means of describing motions. We might go on obtaining still further 

 descriptions by means of higher derivatives of the coordinates, but 

 experience shows us that nothing is gained thereby, for, in the 

 great majority of cases with which we have to deal, it is found that 

 the components, X, Y, Z, are expressible as functions of only the 

 coordinates of the bodies involved, or at most of the coordinates 

 and their first time derivatives. 



There is a further advantage in the introduction of the notion 

 of force, in that if a body be submitted to the action of two agencies 

 at different times, so as to move under the influence of definite 

 forces, and then be submitted to the action of both simultaneously, 

 the force now found to be acting will be the resultant of the two 

 original forces. This statement, that forces are compounded as 

 vectors, being the equivalent of the so-called statement of the 

 parallelogram of forces, is implicitly contained in Newton's second 

 law of motion. 



Under certain circumstances, an agent which would under other 

 conditions cause motion, may cause no motion. We then say that 

 its effect is counteracted by that of some other agent, or otherwise, 

 that the two forces are in equilibrium. According to the third law, 

 the two forces are equal and opposite, either being the reaction 

 with respect to the other. Such reactions are called static reactions, 

 as opposed to the kinetic reactions exerted by bodies undergoing 

 acceleration. 



As has been stated above, most of the forces which occur in 

 nature depend only on the positions of the bodies upon which they 

 act, or at most upon their positions and velocities, but not upon 

 the higher derivatives of the coordinates. Forces of the former sort 



