forces parallel to three rectangular axes ox, oy, oz, fig. 10, which would represent the action of the forces mA, mB, &c., estimated in the direction of the axes; or, which is the same thing, each of the forces mA, mB, &c. acting on m, may be resolved into three other forces parallel to the axes.
33. It is evident that when the partial forces act in the same direction, their sum is the force in that axis; and when some act in one direction, and others in an opposite direction, it is their difference that is to be estimated.
34. Thus any number of forces of any kind are capable of being resolved into other forces, in the direction of two or of three rectangular axes, according as the forces act in the same or in different planes.
35. If a particle of matter remain in a state of equilibrium, though acted upon by any number of forces, and free to move in every direction, the resulting force must be zero.
36. If the material point be in equilibrio on a curved surface, or on a curved line, the resulting force must be perpendicular to the line or surface, otherwise the particle would slide. The line or surface resists the resulting force with an equal and contrary pressure.
37. Let oA=X, oB=Y, oC=Z, fig. 10, be three rectangular component forces, of which om=F is their resulting force. Then, if mA, mB, mC be joined, om=F will be the hypotenuse common to three rectangular triangles, oAm, oBm, and oCm. Let the anglesmoA=a, moB=b, moC=c; then
X=F cos a, Y=F cos b, Z=F cos c.(1).
Thus the partial forces are proportional to the cosines of the angles which their directions make with their resultant. But BQ being a rectangular parallelopiped
F2=X2+Y2+Z2.(2).
Hence
X2+Y2+Z2F2=cos2 a+cos2 b+cos2 c=1.
When the component forces are known, equation (2) will give a value of the resulting force, and equations (1) will determine its direction by the angles a, b, and c; but if the resulting force be given, its resolution into the three component forces X, Y, Z, making
