MECHANICAL PHILOSOPHY. STATICS. 



[FORCES in OODPLM. 



Hence D and E are both in the same straight line. 



Since any fore* may be transferred from its point of 

 Application to any other point in the line of its action, 

 wo may remove the forces P, and P. from the points A 

 and A 7 to the point D, and the forces P, and H, from 

 B and B' to E. For die sake of clearness, the new posi- 

 of the forces will then be represented by the two 

 following figures one representing the positions of the 

 two forces P, and P., 

 which have not had their 

 points of application al- 

 tered, forming a couple, 

 nd acting at the extre- 

 mities of the arm A'B' 

 (Fig. 60) ; aud the other, 

 the positions of the 

 forces whose points of 

 hvn 



Now, completing the 

 parallelognuiis P. 1) 1', 

 Hand E P t R' P, ; D R and E R' their diagonals will 

 represent the resultants of the forces P, and P, acting 

 at D, and P, and P acting *' " 



at K. And since the four 

 forces P. P., P, and P t 

 are equal, as are also the 

 angles at D and E, the 

 forces represented by D R 

 and EH' will be equal ; 

 and since they attain op- 

 posite directions in the 

 same straight line DCE, 

 they will counteract each 



Hence on the whole we hare replaced the couple 

 P -A B by the couple P A' B', without altering the con- 

 ditions of equilibrium. 



PROPOSITION XIIL 



Tkt effect of a couple will not be. altered if it* arm be re- 

 moved to any putitivn parallel to ittelf, either in itt own 

 plane or elte in one parallel to it. 



Firstly, let the arm A B of the couple P A B be re- 

 moved to the new po- 

 sition A' B' (Fig. 52 

 1), A'B' being pa- 

 rallel and equal to 

 A B and in the game 

 plane. 



The conditions of 

 equilibrium will not 

 be altered if we apply 

 ] to A' two forces 1> S 

 f ami P., acting in op- 

 i posite directions per- 

 \ pendicularly to A' B', 

 ', and both equal to 

 ; P, orP t . 



/ Similarly we may 

 apply at B' the forces 

 P, and P.. each equal 

 to the former, and 

 perpendicular to A' B'. 



Join A'B, Af meeting in C. Then since AB 



': 



J 



I! ^ 

 t - 



parallel and equal to A'B', we have in the triangles 

 ABC, A'B'C, the angles at A and B oqual to the 

 angles at B' and A', and also A B - A' B'. Heuce A C 

 = B'CandBC = A'C. 



Now for the sake of clearness we may suppose the six 

 forces represented above divided into two groups, as in 

 Nos. 2 and 3, oue (2) consisting of the arm A' B' acted 

 on by the equal and opposite forces I'., :nnl ]', forming 

 a couple P A' B' ; and the other (3) of the four Mod 

 and parallel forces ; P, and 1', acting at A and I! in 

 one direction, and P 4 and P 5 at A' aud B iu the oppo- 

 site direction. 



Now by Prop. XI., since A C = B'C aud P, - P S) 

 Fig. !-(). 



H ' 



P, and P 5 acting at A and B' will be equivalent to a 

 single force C R, = 2 P acting atC in the direction C R, 

 parallel to AP, or B'P,. Similarly P, and 1\ acting 

 at A' and B will be equivalent to C R <= 2 P acting at 

 C in the direction C R., parallel to A' P 4 or B P,. But 

 the two equal forces CR, ami C l: . :i.-tinx on V in oppo- 

 site directions in the same straight line will counteract 

 each other. 



Hence the group of forces P,. P , P 4 , and P, will 

 neutralise each other, and we shall only have the effect 

 of the couple P' A'B' left, as represented iii the other 

 group (4). 



On the whole, therefore, we have replaced P -A B by 

 the couple P- A'B' the arm A'B' being p:irallel and 

 equal to the arm A B ami in tin- same plane with it. 



Secondly, let the arm A B be removed to the position 

 A'B' parallel to AB, in a plane parallel to the piano of 

 the couple P -AB(4). 



Lot D E H I represent the plane of the couple P A B 

 F<; K L a plane parallel to D I-'. II I, 1 ho plane to which 

 the arm AB is supposed to bo ri'iuo\v<l 



For the sake of clearness we may con-i'i' i thrsr planrs 

 as opposite faces of the parallelepiped D E K< i H I K I . 



Thi-n A' B', the new position of the arm, will ! drawn 

 in tin- plane K C K L, parallel and equal to A B in the 

 plane 1 1 [.ill 



loin A B', B A' meeting in 0, ami apply, as in the 

 procedirg case, equal and opposite forces l' s , 1'^, l' B , 1' 8 



