MECHANICS. 



passing these strings over wheels abed, 

 let weights A B C D be suspended from 

 them. 



Take any part P m on the string P a, 

 and from m on the board to which the 

 apparatus is attached, draw a line pa- 

 rallel to the string P b, and take a part 

 m n upon that parallel, such that P m : 

 m n : : A : B. Again, through n draw 

 a parallel to the string P c, and on that 

 parallel take a part no, such that m n : 

 B : C. In like manner draw op 



parallel to P d, and such that no: op:: 

 C : D. Finally, join the points p and P 

 by a right line. A single force, acting 

 in the direction of the line Pp, and 

 having the same ratio to each of the 

 other forces as the line Pp has to the 

 side of the polygon, which is parallel to 

 that other force, will produce a pressure 

 on the fixed point P equivalent to the 

 combined actions of the forces A B 

 C D. This may be established by the 

 same means as were used in the former 

 case. Let a string, attached to the fixed 

 point P, be carried over a wheel e, so 

 that the string P e shall be the continu- 

 ation of the line p P, and let a weight 

 E be suspended from it which shall 

 have the same proportion to the w : eights 

 A B C D, as the side P p has to the 

 sides of the polygon parallel to the 

 strings respectively. If the point P be 

 then disengaged from the board, it will 

 be found to maintain its position, and 

 remain at rest ; so that the force E in 

 the direction P e, immediately opposite 

 to Pp, counteracts the combined effects 

 of the forces A B C D ; and it would 

 also counteract the effect of a force 

 equal to E, in the direction of the line 

 Pp (5.) Hence such a force is equi- 

 valent to the combined effects of A B 

 C D, and is therefore their resultant. 

 From whence we infer the following 

 general theorem : 



If several forces act on the same point 

 parallel and proportional to all the sides 

 of a polygon taken in order except one, 

 a single force proportional to, and in the 

 direction of, that one side will be their 

 resultant. 



It may be proved that this is the only 

 resultant, in the same manner as in the 

 former case ; for if either the direction 

 of the string P e, or the magnitude of 

 the weight E, or both, be in any way 

 changed, the point P, when disengaged, 

 will no longer maintain its position, but 

 will move until it settles into that posi- 

 tion in which the sides of the corre- 

 sponding polygon are proportional to 

 the weights suspended from the strings 

 to which they are parallel. 



(12.) Hitherto we have supposed that 

 the forces applied to the point are in the 

 same plane. This, however, is not at 

 all necessary, and the same principles 

 exactly will apply when the forces act 

 in different planes. 



(13.) It {'must be evident from the 

 reasoning in the preceding articles, that 

 if any number of forces acting on a 

 point, be proportional to the sides of a 

 polygon which are severally parallel to 

 the direction of the forces, and that the 

 forces are in the directions of the sides, 

 taken successively in the same order, 

 the point will be kept at rest, and the 

 forces will be in equilibrium, and will 

 neutralise each other's effects. 



Since any one side of a triangle or 

 polygon is always less than the sum 

 of all the remaining sides, it follows that 

 a mechanical effect will always be more 

 economically produced by a single force 

 acting in the proper direction, than by 

 a number of forces acting in different 

 directions. 



(14.) All that we have established 

 respecting the composition of forces or 

 pressures, also applies to the composi- 

 tion of motions. Two impulses, which, 

 separately communicated, will cause a 

 body to move over the sides of a paral- 

 lelogram in the same time, would, if 

 communicated at the same instant, 

 cause the body to move over the dia- 

 gonal of the parallelogram in that time. 

 This may be submitted to actual expe- 

 riment. 



Let a level table, A B C D (Jig. 5.), in 

 the form of a parallelogram, be provided, 

 furnished with a ledge to prevent a ball 

 from rolling off; and let tw r o spring guns, 

 G G', be placed at one of the corners A, 

 so that when G strikes the ball X, it shall 

 move along the side A B in a certain 



