DYNAMICS. 



289 



J.aws of 

 Motion. 



tage of reasoning. On the other hand, we would wish 

 _ to borrow PS little from external nature, and as much 



""Y""" * f rom the resources of mind, as possible. Provided the 

 workmanship is good, and the foundation sound, we 

 should consider the edifice of science as the more beau- 

 tiful and not less secure, the loftier its summit and the 

 narrower its base. But at the same time it is highly 

 important to know the grounds of our knowledge, and 

 to mark the point where demonstration must end and 

 observation begin. Such nice discussions about the 

 boundaries of evidence, sharpen the intellectual facul- 

 ties, make us better acquainted with the instruments 

 of knowledge, and thus prepare us for pursuing with 

 greater success the career of future discovery. 



This second law, as well as the first, is perpetually 

 confirmed by experience, and by its success when as- 

 sumed as a principle of philosophy. The case in which 

 the forces are pressures, might be deduced with ion-i- 

 derable confidence from the case of impulses, or con- 

 versly, since we have shewn in the first Section, that 

 pressures are proportional to the motions which they 

 generate. An experiment illustrative of this case will 

 be seen in Case 1. Sect. 3. 



Third Lam, or Lam of Reaction. 



Third law When one body acts on another, the other exerts an 

 of motion, equal force against it in the opposite direction, which is 

 usually expressed by saying, that action and reaction are 

 equal and contrary. 



This law holds whether the bodies attract or repel 

 one another, and whether they act at a distance or in 

 apparent contact ; and the grand basis on which it rests 

 . is experience. 



Thus, if one Ixxly attract another at a distance, the 

 other attracts it with equal force in the opposite direc- 

 tion ; .-o th ;t. it' motion ensues, the quantity of motion 

 produced in the one is equal tci that produced in the 

 other ; and, if motion is prevented, the pressure mv. -- 

 sary to prevent it in the one, is the same as in the 

 other. 



If one body repels another at a distance, the other 

 repels it with equal force in the opposite direction, the 

 equality of force being estimated in the same way as 

 above. 



The same law holds when the bodies act in apparent 

 contact. In this case, if they attract, you have no 

 greater reason to ascribe the force with which they co- 

 here, or which is necessary to sejwrate them, to the at- 

 traction of the one than of the other; if they repel, as 

 in impulse or pressure, the reaction seems to hie inti- 

 mately connected with our first conceptions of matter. 

 When one body strikes against another at rest, without 

 producing motion, the same change is induced on the 

 striking body, as if it being fixed had been struck by a 

 body of equal mass and velocity in the opposite direc- 

 tion ; and if it be the hand that strikes, the same sensa- 

 tion is experienced. These circumstances imply an 

 equal reaction. If two bodies meet when moving in 

 ppoMte directions, with equal momenta or quantities of 

 motion, we may presume they will stop one another ; 

 for, equal momenta being the effects of equal forces, 

 may be supposed to possess equal forces, anil such will 

 balance one another. This is actually the case. It ap- 

 pears from this that a force will be able to destroy just 

 as much motion as it can communicate, and hence that 

 it may be measured by either of these effects. If a l>ody 

 strike another at rest, or moving before it in tjie same 



VOL. vui. PART i. 



Laws of 

 Motion. 



direction, we may expect from the analogy of the last 

 case, that the reaction will be equal to the action. We 

 accordingly find that the quantity of motion which the 

 impinging body has communicated is equal to what it 

 has lost, that is to what the other by its reaction has de- 

 stroyed. If the bodies move in opposite directions, we 

 may presume that the body with the greater momentum 

 will prevail, that it will first spend some of its motion 

 in reducing the other to rest, and then some of the re- 

 mainder in moving the other, and that in both parts of 

 the process the reaction will be equal to the action, ac- 

 cording to two foregoing cases. This is also really the 

 case. The same holds in all analogous cases, when bo- 

 dies press against one another. If the bodies, in con- 

 sequence of meeting, alter then- shape, and again re- 

 cover it, they will act on one another during the re- 

 covery, and then also the action is mutual and equal. 



The same law may be expected to hold, and actually 

 holds, when one body acts on another through the me- 

 dium of a third, as when one body pulls another by 

 means of a thread, or repels it by means of a pole. In 

 this case the third body is merely a medium of commu- 

 nication, and cannot be supposed to alter the action of 

 the forces, unless that its mass when considerable would 

 need to be taken into the account. 



SECTION III. 

 I 



Composition of Forces.- 



A body is often acted on by several forces at once, Composi- 

 and it is required to find the effect of this joint action, tion of 

 or a single force which would produce the same effect. forces - 

 This single force is called the equivalent, resulting, or 

 compound force ; and the process for finding it is called 

 the composition of forces. The whole depends on the 

 second law of motion. 



( ' \ - 1: 1 . 7'uo forces applied at the same point of a body. 

 If they act in the same direction, the equivalent will 

 be in that direction, and equal to their sum ; if in op- 

 posite directions, the equivalent will be in the direction 

 of the greater, and equal to this difference ; if at an 

 angle, the equivalent will be in the same plane, and re- 

 pn-scnted by the diagonal of a parallelogram of which 

 the two sides represent the simple forces. 



This follows immediately from the second law of 

 motion ; the only case which needs any demonstration 

 is that when the forces act at an angle. 



If the one force alone would make the body move p tATE 

 along AH, and the other along AC in the same time, CCXLI. 

 the two by their joint action will make it move along *''g- ! 

 the diagonal AD in the same time. 



For by the second law, its motion par.illel to AB will 

 be equal to AB, and its motion parallel to AC will be 

 equal to AC ; hence at the end of the time it must be 

 at D. In like manner .suppose that by the separate 

 action of the forces, the body would at any intermediate 

 point of time be at E and V, then completing the paral- 

 lelogram FE, their joint action will bring it to G at 

 that time. But it is easy to see that AB : AE : : AC : 

 AF. For both motions being uniform, (first law of mo- 

 tion,) the spaces past over in both cases are proportional 

 to the times, and hence proportional to one another. 

 Hence the two parallelograms are similar, and conse- 

 quently about the same diagonal, (Euc. xxvi. G.) 



If the forces are pressures, the effects of the momen- 

 tary action will be extremely small, but the same de- 

 monstration still applies. Though equilibrium is not 

 SN 



