MECHANICS. 



when a body is drawn in directions 

 immediately opposite by two unequal 

 forces, it is affected exactly in the same 

 manner as if it were drawn by a single 

 force equal to the difference between the 

 two forces, and acting in the direction 

 of the greater force. 



(7.) This single force, whose ac- 

 tion is equivalent to the combined 

 action of two or more forces, is called 

 their resultant: and the process by 

 which a single force equivalent in its 

 effect to two or more other forces is 

 found, is called the composition of force. 



(8.) On the other hand, two or more 

 forces may be found whose combined 

 effects are equivalent to that of a single 

 given force ; the process by which these 

 are determined is called the re-solution 

 of force; and the two or more forces 

 which are equivalent to the single force, 

 are called its components. 



(9.) Having considered the simpler 

 instance in which the directions of the 

 forces are in the same straight line, let 

 us now examine the more complex case 

 in which two forces act on the same 



point in different directions. Let P (fig. 

 3.) be a fixed point to which three strings 

 are attached ; and let the strings Pa 

 and P b be passed over fixed grooved 

 wheels as before, and let any weights A 

 and B be suspended from them. The 

 point P is now drawn by two forces A 

 and B, in the directions P and P b. 

 The question is, what single force would 

 produce the same effect upon it ? Take 

 lengths P m and P n on the strings, so 

 that they shall be in the same propor- 

 tion as the weights A and B, that is, so 

 that P m : P n : : A : B ; and upon the 

 board to which the wheels a, b are sup- 



posed to be attached, draw the parallelo- 

 gram P m o n. Draw the diagonal P o. 

 A single force acting in the direction of 

 the diagonal P o, and having the same 

 ratio to the weight A or B as the dia- 

 gonal P o has to the side P m or P n of 

 the parallelogram, will produce the same 

 pressure on the point P as the combined 

 actions of A and B produced. To prove 

 this, let a third wheel c be so placed 

 that the thread P c shall, when stretched 

 over it, be in a direction immediately 

 opposite to P o, and suspend from it a 

 weight C which shall have the same 

 proportion to A or B as the diagonal P o 

 has to P m or P n. If the "point P, 

 hitherto supposed to be fixed, be dis- 

 engaged and left, free to move, it will be 

 found to maintain its position and re- 

 main at rest. Hence it follows, that the 

 weight C neutralises the effects of A 

 and B, and keeps them in equilibrium. 

 But it would also keep in equilibrium a 

 force equal to C in the direction P o (5.); 

 from whence it follows, that a force 

 equal to C in the direction P o is equiva- 

 lent to the united actions of the forces 

 A and B, in the directions P m and P n. 

 Hence we derive the following import- 

 ant theorem :* 



If two forces acting on the same point 

 in the directions of the sides of a paral- 

 lelogram be proportional in their inten- 

 sities to these sides, their united effects 

 will be equivalent to that of a single 

 force acting on the same point in the, 

 direction of the diagonal of that paral- 

 lelogram, and whose intensity is pro- 

 portional to the diagonal. 



This single force in the direction of 

 the diagonal is therefore their resultant. 



(10.) It will very easily appear that 

 two forces have but one resultant ; for, 

 if the force C be in the least degree 

 altered, either in its magnitude or in its 

 direction, the point P, when disengaged, 

 will no longer maintain its position /but 

 will move until it settles into such a 

 position that the magnitudes of the dia- 

 gonal and sides of the corresponding 

 parallelogram shall be proportional to 

 those of the forces ABC. 



(11.) We can now extend our inves- 

 tigation to the combined action of three 

 or more forces on the same point. Let 

 P, (fig. 4.) as before, be a fixed point to 

 which several strings are attached, and 



* This theorem admits of rigorous demonstration 

 independently of the experimental proof which we 

 have given. The demonstration is, however, of too 

 complex a character, and requiring the aid of ma- 

 thematical reasoning of a kind which we cannot 

 properly introduce here. 



B'2 



