FORCES IX COCTPLES.] 



MECHANICAL PHILOSOPHY. STATICS. 



705 



at A' and B', each equal and parallel to P, or P.,. Then 

 Euc,, B. XI., Prop. II. , the triangles A B C, A' B"' C are in 



librium will not be disturbed by the alteration we have 

 made in the position of their arms. Hence, the parallel 



Fig. 53. 



the same plane, and as in the preceding case A C = 

 B' C" and A' C = B C. (Fig 525). 



The parallel forces P, and H e acting at A and B are 

 equivalent to C R. = 2 P acting at C parallel to P or 

 P 6 ; and the parallel forces P, and P 4 acting at B and A' 

 are equivalent to C R, = 2 P acting at C parallel to Pj 

 or P 4 . The forces C R, and C It, neutralise each other, 

 kaving, as in the previous case, the forces P 3 and P 8 

 acting at A' and B' as a couple P'A' B'. 



Combining the two last proportions, we see that the 

 arm of a couple may be transferred from any position in 

 its own plane, to any other position in that plane, or to 

 anv plane parallel to it. 



Thus, if we wish to transfer the couple P- A B (5) to 

 the position, in which one extremity of its arm shall 

 correspond to the point D, and the arm itself be in the 

 direction D E, all we have to do is to take any point C 

 in A B, and through C draw F G parallel to D E. Then 

 by Prop. XII. , P A B may be transferred to the position 

 P FG, and from that, by Prop. XIII., to the position 

 P D H. F G and D H being both equal to A B. 



PROPOSITION XIV. 



Couple* acting in the tame plane or in planet parallel to 

 each other, will be equal if their moments be equal. 



Let P'AB (Fig. 53 1) be a positive couple, and 

 Q D E (2) a negative one, either acting in the same 

 plane with P AB or else in one parallel to it. P being 

 greater than Q, and D E greater than A B. 



Then the couple Q D E can be transferred so as to have 

 its arm in such a position that one extremity B shall 

 coincide with A (1), and its arm be the same line as A B. 

 is represented in (3), A F being equal to D E. Now, 

 the opposite forces P and Q acting at A, are equivalent 

 to a single force P Q acting at A in the direction A P, 

 as shown in (4). 



Hence the effect of the two opposite couples P A B 

 and Q D E on the rigid body, on which they are sup- 

 posed to act, will be equivalent to the three parallel 

 forces P Q, P and Q acting at the points A, B and F 

 in the same straight line A H F. 



Now, if these couples P'A B and Q-D E, are snch as to 

 neutralise each other or produce equilibrium, the equ> 



VCK* 1. 



forces P Q, P and Q acting in the points A, B and F 

 in the line A B F, must in this case produce equilibrium, 

 and by Prop. XI. we shall have 



(P Q) AB = Q-BF 



or P-A B = Q A B + Q B F = Q (A B -f B F) 

 = QDE; 



which shows that the moments of the two couples are 

 equal. But a negative couple P A B would counteract 

 the positive couple P -A B if their arms were in the same 

 position. 



Hence, two negative couples will be equivalent to each 

 other, if their moments be equal, provided only that 

 they act in the same plane or in planes which are parallel 

 to each other. 



The same reasoning will apply to positive couples, 



PROPOSITION XV. 



To find the resultant couple of two couples, which do not 

 act in the same plane on a rigid body. 



For the sake of clearness, we shall suppose the planes 

 in which the two couples act, represented by the open 

 pages of a book E F G H K L, standing upon a table 

 (Fig. 54 1), and inclined to each other at an angle 0. 

 E H will be the intersection of the two pages or planes, 

 which are both supposed to be perpendicular to the sur- 

 face plane of the table. 



Let P'.A B be the couple acting in the plane LEH K, 

 Q ' C D that acting in the plane E F G H. 



By the previous proposition the couple P'AB may be 

 replaced by the couple P'- E H acting in the same plane 

 L E H K, provided P'- E H = P'A B, 



AB 



or, P' . 



EH 



AB. 

 InHKtakeHP'=P-.gjj 



Produce L E to P' and make E P' =H P'. 



Then P' E H P' will represent in magnitude and 

 direction the couple P''E H, which replaces P'AB 



4 x 



