MECHANICAL PHILOSOPHY. STATICS. [TBIANOLB, ETC., OF FOBCIS. 



Lastly, draw P 4 R, parallel to AR, and R, R, 

 parallel to A P 4 , meeting in R s . Join A Rs. Then by 

 Prop. I. a force R,, represented in magnitude and 

 direction by A R,, will have the same effect on A as th o 

 two forces R, and P have, acting in tho directions ARi 

 and A P,. 



But R,, acting on A in direction AR 2 , produces the 

 same effect as P, P s and P s acting in tho directions AP,, 

 AP 2 , andAP,. 



Consequently a force R 3) acting in the direction AKa, 

 produces on A the same effect as the four forces P,, 1 3, 

 P s , and P 4 , acting in tho directions AP,, A P., AP 3 , 

 and A P. ; or, in other words, A R 3 represents tho re- 

 sultant of these forces in magnitude and direction. 



The same method may be extended to any number of 

 forces, and affords an easy geometrical construction for 

 finding the single resultant of any number of forces 

 acting upon a material particle. 



PROPOSITION III. 

 Resolution of Forces. 



By means of the parallelogram of forces wo can gene- 

 rally replace a single force by two others acting in any 

 directions wo please in the same plane ; this is called re- 

 solving a force, and the forces by which it is replaced are 

 termed its resolved parts. 



Thus if a force P (Fig. 24) acting on a point A, be 

 represented in magnitude and direction by the straight 

 line A P ; and A B, Fig. 24. 

 A C drawn through c < 



Abe the arbitrary 

 directions in which 

 we wish to resolve 

 the force P. t- -/e 



Through P draw 

 P D parallel to A C 

 meeting A B in D, 

 and also PE parallel 

 to A D meeting A C 

 in E. 



Then, by Prop. I., AE and AD will represent two 

 forces in magnitude acting in the directions A C and A B, 

 which produce on A the same effect as the simple force 

 P acting in the direction A P, and therefore may replace 

 that force without altering the conditions of equilibrium. 



A E represents the resolved part of P along A C, and 

 AD its resolved part in the direction A B. 



In resolving forces, it is generally found more con- 

 venient to choose the direction A C perpendicular to A B. 



PROPOSITION IV. 



Triangle of Forces. 



If a material point be kept in equilibrium by the action of 

 three forces acting upon it, in the same plane, the sides 

 of any triangle drawn parallel to the directions of these 

 three forces will be projxtrtional to them ; and conversely, 

 three forces, acting on a material particle, will keep it at 

 rest if tliese forces be proportional to the sides of a 

 triangle' formed by drawing lines parallel to their 

 directions. 



Let P and Q (Fig. 25), two forces acting on A, be 

 represented in magnitude and direction by the hues A f 

 and A Q. 



Through P draw 

 PB parallel to A Q, 

 and through Q,Q B 

 parallel to AP meet- 

 ing in B ; join A B. 



Produce A B to R, 

 and make AR equal 

 to AB. 



Then Prop. I., a 

 force R acting on A, 

 represented in mag- 

 nitude and direction 

 by AR, will counter- 

 act the forces P and 

 Q acting in the direc- 

 tions A F and A Q. 



And A will bo kept in equilibrium by tho action of 

 the throe forces P, Q, and R acting in tho directions 

 A P, A U, and A R. 



Take any point E ; through E draw E D parallel to 

 A P, and E F parallel to A R. 



In E D take any point D, and through D draw D F 

 parallel to A Q, meeting E F in F. 



Then by construction tho triangle E D F will bo equi- 

 angular to the triangles A P B or A U It. 



And, by Euc. B. VI., Prop. 4, ED : EF : FD : : 

 AP:AB:BP::P:R:Q. 



PROPOSITION V. 



Polygon of Forces. 



If a particle be acted on by any number of forces whidi are 

 r presented in magnitude, or are proportional to the 

 sides of a polygon, it will be at rest, prowled each Jorce 

 acts in a direction parallel to tlie side of the polygon to 

 which it is proportional. 



Let A B C D E F (Fig. 2G) be a polygon, whose side 

 A I!, is proportional to the force P lf B C to P. 2) CD 

 to P 3 , D E to P 4 , E F to P B , 

 and K A to P B . 



Join AC, AD, and A E. 



Let AC = R!, AD=R 2 , 

 and A E = R 3 . 



Then, by Prop. IV, R, 

 will represent the resultant 



Fig 20. 



"(- BZ 



of the "forces P r and P 2 in 

 magnitude. 



R., the resultant of R, ar.d 

 P,, or of P., P., and P 3 . 



And R, the resultant of R 2 , and P t , or of P,, P a 

 P,, and P,. 



"But, by Prop. IV., since R s , P 6 , and P 6 are sides of 

 the triangle AEF, R 3 will represent the resultant 

 of P. and P 6 . 



Hence P. and P 8 will counteract the effect of the 

 forces P., Po, P, and P o provided the conditions of 

 Prop. IV. be fultilled ; i.e., provided these forces act 

 upon a particle, in directions parallel to the sides of tho 

 polygon. 



Since, in Prop. IV., any side of the triangle may bo 

 considered as the resultant of the other two, !' may bo 

 regarded as the resultant of R 3) and P,, or of P,, P u , 

 P 3 , P., and P K . Anyone side, therefore, of tho poly- 

 gon may be taken as the resultant of all tho oil 



In the same manner the proposition may be extended 

 to a polygon of seven, eight, or more sides. 



The polygon of forces need not have all its sides in the 

 same plane. 



PROPOSITION VL 

 Pardllelopiped of Forces. 



If three forces, wlwse directions are not in the same plan f, 

 act upon a point, and if they be represent 

 tude and direction by three adjacent edges of a}"* 

 wiped, which meet in the point on which th 

 tlie resultant of the forces will be re } >resented in 

 tude and direction by the diagonal drawn fro,,, (Ml 

 point to the opposite solid angle of the par,' 

 Let A bo the point (Fig. 27), acted on by three forces, 

 represented in magnitude and Fig. 27. 



direction by AP, AQ, and 

 AR, which are not in the 

 same plane. 



Then, Euc. B. xi. Prop. 2, 

 A Q and A P are in the samo 

 piano ; complete the parallelo- 

 gram PAQB in this plane, 

 jpy drawing PB parallel to 



A Similarly in' tho plane in 'which AR and AQ Ho 



AP lie, complete 

 grams QCBS in the plane 



