MECHANICAL PHILOSOPHY. 8TA.TIO& [FAKUUUUI OFFOV 



extremity, c*ro being taken to support this latU-r weiglit 

 so as not to allow it to act on A till wo require it. If, 

 now, the weights of 2 IK and 3lb. bo supported so as to 

 take off their tensions from A at the same instant that 

 the second 4 II allowed to exert its tension on 



A in \ !', the equilibrium of the point A 



will not be dixturlx*), since the tension of 41bs. acting iu 

 the direction A E, will exactly balance the tension of the 

 41b. net in in tho dinvtion AC. The tension of Alb. 

 acting in the direction A K. produces, therefore, the same 

 "on the point A that the joint tensions of 3 Ib. acting 

 in the direction A B, and of 21b. in the direction AD, 

 both together have upon it. The 4 Ib. tension, acting in 

 direction A E, is called tho resultant of tlio 3 11). 

 tension in the direction A B, and the 2 Ib. tension in that 

 of A D. 



__ _ PROPOSITION I. 



1'AUALLELOGRAM OF FORCES. Tlio proposition 

 which enables us to represent the resultant of any two 

 forces which act upon a material particle in magnitude 

 and direction, when tho magnitude and direction of the 

 two forces are given, is called the parallelogram of forces, 

 and is as follows : 



Let P and Q be two forces 

 acting upon material point A. 

 (Fig. 12). 



Let the line A B represent 

 the force P in magnitude and 

 direction, and A C the force Q 

 in magnitude and direction. 



Through B draw B D parallel 

 to A C, and through C, C D parallel to A B. 



Lot 1) be the point where the lines BD and CD 

 Mb 



Then, by construction, the figure A B D C is a parallel- 

 ogram. 



Join AD, A D is the diagonal of the parallelogram. 

 This diagonal A D will represent the resultant R of the 

 two forces P and Q acting in the directions A B and A C. 

 in magnitude and direction. 



The parallelogram of forces may therefore be thus 

 enunciated. If two forces acting upon a material particle 

 be represented in magnitude and direction by two ad, 

 fides of a parallelogram, the diagonal of the parallelogram, 

 drawn through the point where tltcsc sides meet, trifi 

 represent their resultant both in magnitude and dine- 

 tion. 



We shall first show that this proposition is true for the 

 direction of tho resultant, and then that it is also true for 

 its magnitude. 



1st. To prove that the parallelogram of forces is true for 

 the direction of tho resultant. 



When the forces P and Q are equal, 

 ^ the direction of the resultant will mani- 

 festly bisect the angle B A C (Fig. 13), 

 since no reason can bo alleged why the 

 resultant force R should incline more 

 to one force, P, than to the other, Q. 

 Since A B CD is a parallelogram whose 

 sides AC and AB aro equal, its diagonal, 

 A D, will bisect the angle B A C, and therefore A D will 

 represent the direction of the resultant when P and Q are 

 equal. 



Let us now assume the proposition to be truo for two 

 unequal forces P and Fi - 14 - 



Q, and also for two 

 unequal forces P and 

 S (Fig. 14), wo can 

 then prove that it 

 will bo true for the 

 forces PandQ-f 8. .. 



Draw a parallel- 

 ogram A B D C, having one dido A B proportional to tho 

 force P, and tho adjacent side A C to that of Q ; produce 

 A C to E, and make C E in the same proportion to S tliat 

 the other lines bear to P and Q. Complete tho parallel- 

 ogram ED. 

 According to our assumption, the resultant of P and Q 



will act in tho direction A D, and that of P and S in tho 



direction C F. Now 

 if all tho points in the 

 two parallelograms bo 

 supposed to be rigidly 

 connected with one 

 another, a force may 

 be transferred from 

 any point to another, 

 provided the latter be in the same straight line in which 

 tho force is acting, without disturbing the equilibrium. 



Hence tho force S acting at C may be transferred to A 

 (Fig. 15), and wo shall then have a force Q + S acting on 

 A in the direction A C, a force P actin<j on A in the di- 

 rection A B, and a force P acting on C in the direction 

 CD. 



If the proposition be true for P and Q + S, their 

 resultant will act in 

 the direction A F, 

 and these forces 

 may be transferred 

 from A to F. What 

 wo have therefore 

 to show is, that tho 

 forces P and Q act- 

 ing at A, and P and 

 S acting at C, may be so transferred, by the principle of 

 transmission of force, without altering tho conditions of 

 equilibrium, that we may have the forces Q + S and P 

 acting at F and P at C (Fig. 1C). 



By our assumption, P and Q acting at -A may be 

 replaced by their 

 resultant R (Fig. 

 17) acting in the 

 diagonal A D of tho 

 parallelogram C B, 

 and the forces P 

 and S by their 

 resultant T acting 

 in the diagonal 

 C F of tho parallelogram E D. 



Now, according to tho principle of the transmission of 

 force, the force R maybe transferred from A to D in the 

 direction AD, and the force T from C to F iu tho direc- 

 tion C F, as represented in the accompanying diagram 



(Fig. 18). 



Fig. 18. 



Upon tho same principle that wo replaced the forces 



by their resultants, 

 we may replace 

 these resultants 

 again by the f i irees 

 from which they 

 were obtained ; 

 and we shall then 

 have R replaced 

 by the force 1' act- 

 ing in tho direction 

 of C D produced, 

 and Q in that of D F, and T replaced by P acting in E F 

 produced, and S acting in D F produced (Fig. J!h. 



Lastly, tho force Q may be transferred from D to F in 

 tho direction of D F produced (Fig. 20), and the force I' 

 from D to C in the direction CD; so that we ultimately 

 have tho forces Q + S and P acting at F, and 1' at 0, 

 without having altered the conditions of equilibrium of 

 any of tho points of tho parallelograms. Hence, if our 

 proposition be true for two forces P and Q, and also for 

 P and S, as regards tho direction of tho resultant, it 

 is also true for the forces P and Q + & 



