PARALLEL FORCES. ] 



MECHANICAL PH1LOSOPHY.-STATICS. 



707 



This is evidently equal to a single couple whose arm 

 is C D, with equal and opposite forces Q t + Q 3 + Qa 

 Q 4 acting at each extremity. (Fig. 55 2). 



Fig. 55 -(2). 



Hence we have 



Qj 1 - C D Q 4 C D * 



= P! -A! B. + P 2 -A 2 



B 2 + P '-A.., B. P 4 -A 4 B 4 , or the resultant couple 

 is the algebraical sum of the moments of the component 

 couples. 



The same reasoning may be extended from four to any 

 number of couples, and, by Prop. XIII., to couples acting 

 in parallel planes. 



THEORY OF COUPLES. The propositions from 12 

 to 16 contain the fundamental principles of what has 

 been called the theory of couples. For this beautiful 

 theory, which was introduced into the science of Me- 

 chanics about forty years since, we are indebted to the 

 nguished mathematician, M. Poinsot. After we 

 have extended our llth proposition from two to any 

 number of parallel forces, we shall again return to the 

 theory of couples, and determine, by its aid, the condi- 

 tions of equilibrium for any number of forces acting on 

 a rigid body. 



PROPOSITION XVII. 



To find the magnitude and direction of the resultant of 

 any number of parallel forces acting on a rigid body in 

 the same plane. 



Let four parallel forces, P,, P 2 , P 3 , and P 4 (Fig. 56), 

 acting in the name plane on the points A,, A 2 , A., and 

 A 4 , be represented in magnitude and direction by A. 

 P,, A. P 2 , A, P 3 , and A 4 P 4 . 



Join A, A 2 ; then by Prop. XL, B, Q, = Pj + P 2 

 applied at a point B, in A 1 A 2 such that 



A,B, 



A A 



Aj A 3 

 2 



Fig. 56. 



Through B 3 draw P 3 Q 3 = Q 2 + P 4 = P, + P, + P 3 



P 4 parallel to A 4 P 4 . 



Then B 3 Q 3 will represent Q 3 the resultant of the 

 forces Q and P 4 , or of the four parallel forces P,, P 2 , 

 P 3 , andlP 4 in magnitude and direction. 



The same reasoning may be extended to any number 

 of parallel forces. 



It is sometimes far more convenient to refer these 

 various points, A 1 , A 2 , A S) A 4) B,, B 2 , B 3 , <tc., to two 

 fixed arbitrary lines or axes drawn at right angles to 

 each other, as in Props. VIII. and IX. 



Let O X and O Y (Fig. 57), drawn through the point 

 O at right angles to each other, be chosen as arbitrary 

 rectangular axes, to which the points A,, A 2 , &c., B,, 

 B 2 , ,&c., are to be referred. 



Fiff. 57. 



and drawn parallel to A, P, or A 2 P 2 will represent Q, 

 the resultant of the parallel forces P, and P 2 in magni- 

 tude and direction. 



Then join B, A s , in B t A 3 take a point B 2 such that 



Through B, draw B. Q, parallel to A, P, and = P, 



+ P. + P,. 



B 2 Qj will represent Q 2 the resultant of P 3 and Q,, 

 or of 1 J 3 , P a , and P , in magnitude and direction. 



Again, join B 2 \ in B 2 A,, take a point 1'., such 

 that 



Through A,, B,, and A 3 draw A t M,, B, N,, and 

 A 2 M 2 perpendicular to O X. 



Then the lines O M, and M, A,, which determine the 

 position of the point A, with respect to the axes OX 

 and O Y, are called the rectangular co- 

 ordinates of the point A,. 



Similarly, O N t and N t B, are the 

 rectangular co-ordinates of B, ; and O M 2 

 and M 2 A, are those of the point A 2 . 



Let O M, be represented by the sym- 

 . bol x,, OM 2 by x z , A, M, byj/,, and 

 \ A, Mj by y,. 



Through A, draw A, R parallel to O X, 

 cutting B,N, in S and A 2 M 2 in R. 



Then from the construction of the 

 figure, it is evident that the angles at S, 

 K, M, , N, and M, are right angles ; con- 

 sequently Aj S = 5I 1 N 1; A,R = M, M 2 , 

 and also A. M, , S N , , and R M 2 are 

 equal to each other. 

 Now, by the previous part of the proposition, the 

 point B,, was so taken in A l A 2 that 



A i B i 



' A 





i i p-'p; "> AP+P, 1 



Aain, because A 2 R is parallel to B t S, therefore Euc., 

 B. VI., P. II. 



AjJB, A, S^B) S, 

 A, A 2 = A,R A,U 



t S_M, N, O N,-OM 1 

 ~~- 



