MK'M! \MC\1. rilll.M-ol'IIY. ST \TICS. 



[rOKCES IK COV PI.ES. 



A, 8 A, B, I'. 

 HMO. aux X^-ATA-.-pT+T? 



-r, 

 we hv ' ' K- J_TT 



*1 J 1 *!"+" 



Or (O X. x.) (P, + P.) - P. *, P. 

 And ON, (P, + P,)-P, *, -P,*,- i 

 Hence O N, (P, + P.) - P, ^ + Pi *.- 



n _ i _ 



Aud 



p . p 



B, 8 B,N, >N, B.N.- 

 Aguu ' A, K ~ A ~ AjMji- 



..B.fr.-y. 



B.S A.B. P, 



BI A,K A, A, FT+17 



Bj_N,-y,- I' 



" "- 



And(B v N A -y,)(I', + P s ) - 

 -- 



If, now, from B^ and A. we draw B 2 N, and \ 

 A MI perpendicular to X, and represent OM, > ( 

 by i s and A,M, by y,. 



Then, by a similar construction and domonstra- { 

 tion to tliat used for the points A,, Bj and B t , \ 

 we can show that 



But it has Iwen shown that 



But B, N, _ P iV. +PV', or B, N, (P, + P 2 ) - 



*! T r t 



P, y, + P, y s , and substituting this value in the above 

 equation, anil transposing, 'we nave B, N s (Pj -(- P t -f- 



N .. P yi+ P y+Py a 

 1-,+^+p, 



In a similar manner it may be shown that 



+ P. + P 8 



If, now, perpendiculars B., N s and A . M t are sup- 

 posed to be drawn from H s and A t to O X, 



And () M. r 4 , and M 4 A 4 y,, we shall have by the 

 carrying out the same method of demonstration 



''P.+P'. + P. + P* ' 

 And O N. - ?' *>+ ^^ + J.'-- J + l> * x . 



We might proceed from the case of four forces to five, 

 from five_to six, and so on ; so that, assuming the 

 symbols z and y to represent the rectangular co-ordi- 

 ot the point of application of the resultant of n 



parallel force* P., P., P., Ac.. P. referred to thl 

 arbitrary axe. O X ami O Y j we shall have 



If any of the force* act in an opposite .i > the 



othen, they must be takuii with tlio m-^ , ; tin- 



co-ordinates of the various points of ai>]>lk-*ti<>a of the 

 forces must also be taken with the : 

 mined by their ixition with regard to tho axes. 



PROPOSITION' XVIII. 



To find the magnitude and dirtction of V\e ntuUant font 

 aiui rttultunt conyle of any number of force* acting on a 

 ritrid body in the tame plane, and the condition* 'n 

 which there will be equilibrium. 



Let P,, P,, P., and P. (Fig. 681), be four force* 

 represented in magnitude and direction by A, P,, 



nt.n-0). 



A 2 PS, Ai P J; and A t P ; acting on a rigid body in the 

 same plane. At, A ; , An, and A,, being tho points of 

 application of these forces to the rigid body. In the 

 plane in which the forces act, take any two straight lines 

 OX, O Y, at right angles to each other as rectangular 

 axes. 



Let OM t = x,, A, M, = y,, be the rectangular co- 

 ord i nates of AI, referred to the axes OX and (>\ . 



O M 2 = x t , Aj M 3 = y a , those of A 3 ; O M s = x 3 , 

 A, M s yi those of A s ; and O M = x t , A M, = y 4 , 

 those of A,. 



Also let a,, a, a,, and n be the angles that A, PI, 

 A, P 3 , A, Pj, and A P 4 , make with lines -drawn 

 through A,, A,, A 3 , and A 4 parallel to O X. 



For the sake of clearness wo will first confine our 

 attention to the force P,, acting at the point AI. 



Through O (2) draw O a t at right angles to Pi A, 

 produced. 



Also through O draw P/ P./' parallel to A, P,. 



Fig. 58-(2). 



Make P,' and P," both equal to A , P, 



