OF Fences.] MECHANICAL PHILOSOPHY. STATICS. 



C97 



QCB, and the parallelogram PBSD in the plan 

 DPB. 



We shall then have constructed a parallelepiped 

 three of whose adjacent edges meeting at A are A P 

 A Q, and A K 



Join A B, A S, and R S. A B will be the diagona 

 of the parallelogram A P 1! Q, and A S the diagonal a 

 the parallelogram A K S B. 



Hence A B will represent the resultant of the forces 

 repruaeuto 1 by A P and Ay in magnitude and direction 



And AS the resultant of the forces represented b 

 A Rand All. 



Therefore A S represents the resultants of the forces 

 represented by All, AP, and Ay, in magnitude i 

 direction. 



By the aid of this proposition, any single force actinj 

 on a particle may be resolved into three other forces 

 not acting in the same plane. When a force is so 

 resolved, it is generally found convenient to resolve i 

 into three forces acting at right angles or perpendicular!; 

 to one another. 



PROPOSITION VIL 



Condition of equilibrium when a material particle it octet 

 on by any number of force*, ichote direction* art all in 

 the tame straight line, 



The condition of equilibrium in this case must evi- 

 dently be, that the sum of all the forces acting on the 

 particle in one direction must be equal to the sum of al 

 the forces acting in the opposite direction. 



It i* usual to consider the forces acting in one direc- 

 tion as positive, and those in the opposite as negative. 

 The condition of equilibrium, then, may be thus stated : 

 the algebraical sum of all the forces is zero, or nothing. 



Thus, if A be acted on by the positive forces, 41bs., 

 Sib*., and 6lbs. , in one direction, and the negative 

 force*, 61ba., 61bs., and 1 lb., in the opposite, 



The algebraical sum of these force* will be 4 + 3 -f- 6 

 -6-6-1-12-12-0; and A will be at rest under 

 the influence of these force*. 



PROPOSITION VIEL 



// tico ttravjht line* be drawn at riyht angle* to one another 

 throwih any point, and if two force* act on thi* point in 

 any direction whatever in the tamt plane in which thete 

 line* are drawn, then the algebraical turn of the retolved 

 part* of the two force* in the direction of each of the lint* 

 trill be equal to the retolved part of their retultant in the 

 >, direction. 



. 



Let Ax and Ay (Fig. 29) be the straight lines drawn 

 through the point A perpendicular to one another. 



Lines so drawn are called recttitujular axe*. 



Let two forces, P and y, 

 acting on A, be repre- 

 sented in magnitude and 

 direction by A P and A y. 



Draw PR parallel to 

 AQ, and y R to AP, 

 Meting in R, and join 

 AR. 



Then, Prop. I., AR 

 R represents the resultant I 

 of P and <J in magnitude ._ L 

 and direction. *' '* 



Through the points P, y, and R, draw PX,, y X 4 , 

 and R X., perpendicular t through the same 



points P Y,, y Y 2 , and R Y 3 , j trp -ndicul.-u- to Ay. 



By Prop. III., AX, and AY, will be the resolved 

 parts of the force P in the directions Az and Ay. 



Similarly, A X 3 and A Y 3 will be the resolved parts 

 of y ; and A X s and A Y s the resolved parts of R in 

 the same directions. 



Let AX,-X r , AX.-X,, AX,-X 3 , AY,,-Y,, 

 AY.-Y,, and AY, = Y S . 



nee the lino Y a y to meet the lino It X 3 in the 

 point I'.. 



YOL. L 



Since A P R y is a parallelogram, A P R y ; and 

 also because B Q is parallel to AJ:, au J A P to R y, the 

 angle R y B= angle PAX. 



But the angles R By and P X, A are right angles by 

 construction. 



lU-uce, the triangles RyB and PAX, have the side 

 AP = Iiy, and the angles PAX 1 =RyB, and 

 PX, A = liyB. 



Therefore, Euc , B. L, Prop. XXVL, AX,=By, and 

 PX= l: i:. 



And X, + X,, = AX, -r-^AXj = By + AX a 

 =* Xj X 2 ~\- A X 2 = A X a X., . smcii B y = X 3 X.., 

 because by construction of figure B y X., X., is a 

 parallelogram. 



And again Y, -f- Y 2 = A Y, + A Y, = PX. + B X, 



T> T> _T "U %- T> "C A V V 



^ I\ li ^- D Aj ^ Jv Aj ^ A X * I j. 



Since AY, = PX,, A Y z - BX 3 , and RX, = A Y 3 , 

 because AX, PY,, A X 3 BY,, and AX 3 RY 3 , are 

 parallelograms by construction of figure. 



If, therefore, X, Y, be the resolved parts of a force, 

 P, along the rectangular axes, A x and A y, X., Y 2 the 

 resolved parts of y, and X s Y 3 the resolved parts of R, 

 the resultant of P and y in the direction of the sauio 



Fig. 30. 



X 3 - X, + X,, and Y 3 - Y, .+ T,. 



The last figure has been so drawn that the forces P 

 and Q both fall within the axes A z and A y ; this is not 

 always the case ; it may sometimes happen, as in the 

 following tigure, that one of the axes may fall between 

 the two forces ; in this case it will be seen that the 

 resolved part of the force Q, along the axis A y, will 

 fall on the opposite side of A, from the resolved forces 

 of P and R along the same line. 



The resolved forces which lie on one side of A are 

 :allud positive, and 

 thrice on the opposite 

 side negative. 



Consequently, the re- 

 solved parts of P (Fig. 

 30) will boX, an.l V,, 

 of y, X and -Y,, and 

 A : K, X s and Y 8 . The 

 same construction being 

 made as in the last case, 

 since APRyisaparal- X 

 lelogram, A P - R y, 

 and since A X, is paral- 

 lel to B y, and A P to 

 the angle PAX, 



YJ 



, 



= angle Ry B, also the angles at B and X, are right 



Hence, in triangles A P X, and y BR (Euc., B. I., 

 Prop. XXVL), RB = PX,, andBQ-AX.. 



X,+ X^AXj-l- AX a -b y + y Y a = rf Y 2 = X 3 A 



, S -Y S = AY JI -AY 1 |-PX 1 -X 3 B=RB-X, B- 

 1 



Ami Ln this case XjX, + X 3 , and Y 3 =Y, Y.,. 

 A similar construction and demonstration will apply 

 o every other position in which the rectangular axes 

 may be plrccd with respect to the forces, and wo may 

 say, generally, that the algebraical sum of the resolved 

 wrts of any two forces acting upon a material point in 

 he directions of any two rectangular axes, passing 

 hrough that point, will be equal to the resolved parts 

 >i their resultant, along the same axes. 



PROPOSITION IX. 



"ofind the condition* of equilibrium of any number of 

 force* acting upon a material point, the direct loin of 

 the force* being all in the tame plane, but not in the 

 tame ttraiylU Une. 



Let P,, P,, P s , and P 4 (Fig. 31), be four forces 

 acting upon A, represented in magnitude and direction 



y tin- lines AP,, A Pj, A P s , and A 1> 4 . 

 Through A draw the rectangular axes Az and A y. 

 Through the points P,, P 2 , P 3 , and P 4 , draw P t Y t , 



4 c 



