BESTOTANTS OF FORCES.] MECHANICAL PHILOSOPHYSTATICS. 



695 



Now, we have shown that it is true for two equal 



iR. 20. 



CM-S" 



forces ; hence it is tme for P and P, and also for P and 

 P, it must therefore bo true for P and P -f- P> or for P 

 and 2 P. Being true for P and 2 P, and also for P and 

 P, it is true for P and P -f 2 P, or for P and 3 P, and 

 so on ; it may be shown to be true for P and m P, where 

 m represents any whole number. 



By similar reasoning the proposition may be extended 

 to the forces n P and m P, where m and n represent any 

 whole numbers whatever. 



Hence the proposition is true, as to the direction of the 

 result, for any two forces which are commensurable ; or 

 in other words, for any two forces which have a common 

 measure, or can be expressed in terms of a common unit. 



The proposition can be extended to the case of in- 

 commensurable forces, or forces which have no common 

 measure. 



Let the lines AB and AC (Fig. 21) represent two in- 

 commensurable for- 

 ces, in magnitude 

 and direction, of 

 which AC is the 

 groater. Complete 

 th.! parallelogram 

 ABDC, by draw- 

 ing BD parallel to 

 A C, and C D paral- 

 lel to A B. If tho 

 resultant do not act in the direction of the diagonal A D 

 of the parallelogram ABDC, let AE represent the 

 direction in which it acts. 



Divide tho line A C in two equal parts in the point F ; 

 similarly divide F C in two equal parts in G, and G C in 

 H ; continue this subdivision until a part, such as H C, 

 is obtained, which is less than D E. 



A C may, therefore, be divided into a number of equal 

 parts, each of which are equal to H C. 



Set oft' distances each equal to H C along the line C D, 

 commencing from the point C ; then one of these 

 divisions, such as K, must fall between E and D, since 

 A C and C D have no common measure, and H C is less 

 than ED. 



From A B cut off A L= C K, and join K L and A K. 



A C and A L will therefore have a common measure 

 H C, and will consequently represent two commensurable 

 forces ; and A K, the diagonal of the parallelogram 

 A L K. C, will, by what we have previously proved, re- 

 present the direction of their resultant. The resultant 

 A K, therefore, of tho forces A L and A C, is further 

 from A C than A E, the resultant of the forces A C and 

 A B ; but this cannot be true, since AB, being greater 

 than A L, A E ought to be further from A C, or nearer 

 to A B than A K. Consequently, the supposition that 

 tho resultant of A C and A B acts in the direction of the 

 line A E, leads to an absurdity ; and similarly it may be 

 shown, that if it be supposed to act in any other direction 

 than the line AD, the diagonal of the parallelogram 

 A B 1) C, we shall be led to a like absurd conclusion. 



Hence we infer, that if two forces, acting on a point, 

 Fig. 22. whether commensurable or incom- 



mensurable, be represented in mag- 

 nitude and direction by two ad- 

 jacent sides of a parallelogram, the 

 diagonal of the parallelogram will 

 represent the direction in which 

 tho resultant of these forces will 

 act. We have next to show, that 

 this diagonal will also represent the 

 magnitude as well as the direction 

 of the resultant. 



Let A B, one side of the paral- 



lelogram A B C D (Fig. 22), represent a force P, in mag- 

 nitude and direction, and the side A C, another force Q, 

 in magnitude and direction, these two forces both act-ug 

 on the point A. 



Let R represent the force which is the resultant of the 

 forces P and Q ; this force will act, according to what 

 we have already proved, in the direction A D, A D being 

 the diagonal of the parallelogram A B C D. 



Produce AD to D', and take AD' in the same propor- 

 tion to R that A C bears to Q, or A B to P. 



Draw D' C' parallel to A B, and B C' parallel to A I/, 

 meeting in C', and join A C', A B C' D' will be a parallelo- 

 gram, A C' its diagonal, and B C' will be = A D'. 



Since R is the resultant of the forces P and Q, a force 

 R acting oil A in the direction A D', in the same straight 

 line, but in the opposite direction to that in which the 

 resultant of P and Q acts, will keep the point A at rest, 

 when acted on by the forces P and Q. 



Hence the forces P Q and R, represented in magnitude 

 and direction by A B, A C and A D' acting on A, will 

 keep it at rest. 



Any one of these three forces will bo equal in magni- 

 tude to the resultant of the other two, but it will act in 

 an opposite direction to it in tho same straight line. 



Now A C', the diagonal of the parallelogram A 1)' C' B 

 is tho direction in which the resultant of the forces R 

 and P represented by A D' and A li acts. Therefore A C' 

 and A C must bo in the same straight line ; and since AC 

 is parallel to B D, A C' will also bo parallel to B D ; and 

 since B C' was drawn parallel to A 1 ), A C' B D must be 

 a parallelogram, and B C' will = A D. 



But B C' = A D'. Therefore A D = A D'; and since 

 A D' represents R in maynitade, A D will represent R 

 both in m;/ 



We are indebted to 31. Duchayla, a French mathema- 

 tician, for this very simple and beautiful demonstration 

 of tho parallelogram of forces. It may be proved by 

 other methods, but they either require a knowledge of 

 the higher branches of mathematical analysis, or else 

 assume the principles of Dynamics. 



Some writers first demonstrate the properties of the 

 lever, and from these deduce tho parallelogram of forces. 



PROPOSITION II. 



To find the Resultant of any number of Forces acting on a 

 Material Point in the same Plane. 



Let P, P 2 P 3 and P 4 , four forces acting on a point A 

 in the same plane, be represented 

 in magnitude and direction by tho 

 lines A P., AP 2 , A P 8 , and 

 A P i( (Fig. 23). 



DrawP, R t parallel to AP 2 , 

 and P 2 R! parallel to A P, , meet- 

 ing in R, ; join AR^ Then 

 A R t the diagonal of the paral- 

 lelogram A 1\ R t P 8 will byi 

 Prop. I. represent the resultant Us 

 of the forces P, and P 2 acting on 

 A in the directions A PI and AP 2 . 



Let RI be this force. Then R, alone acting on A in 

 the direction A Rj will produce on A the same effect as 

 the two forces P t and P acting together, in the directions 

 AP t andAP 2 . 



Consequently the two forces P! and Pj> acting on A 

 in the directions A P, and A P 2 may be replaced by a 

 single force Ri, acting in the direction ARj. 



Again, draw R t R 2 parallel to A P D and P., R 2 parallel 

 to A Rt meeting in R 2 ; join A R 2 . Then by Prop. I., 

 a force R 2 represented in magnitude and direction by 

 AR S the diagonal of the parallelogram A R! R* P 3 , will 

 have tho same effect on A as the two forces Ri and P 3 

 acting in tho direction A 11, and A P. 



But R! acting on A in the directions A R t produces 

 on A the same effect as Pi and P 3 acting in the directions 

 AP,and AP = . 



Hence the force R 5 acting on A in the direction A R a 

 produces the same effect as the three forces PI P a and 

 P 3 acting in the directions A P^ A P 2 , and A Pa. 



