braical sum of the magnitudes of all the forces taken with their proper signs is nothing. II. The magnitude of the resultant of any combination of parallel forces is the algebraical sum of the magnitudes of the forces. The relations amongst the positions of the lines of action of balanced parallel forces remain to be investigated; and in this inquiry, all pairs of directly opposed equal forces may be left out of consideration; for each such pair is independently balanced whatsoever its position may be; so that the question in each case is to be solved by means of the theory of couples. Prin 39. Equilibrium of Three Parallel Forces in One Plane. ciple of the Lever.-THEOREM. If three parallel forces applied to one A с B Fig. 8. body balance each other, they couples having the same axis, and of equal moments, FA FALA = FB LB, Let according to the notation already used; and let those couples be so applied to the body that the lines of action of two of these forces, FB, which act in the same direction, shall coincide. Then those two forces are equivalent to the single middle force - (FA+F), equal and opposite to the sum of the extreme forces F + FB, and in the same plane with them; and if the straight line ACB be drawn perpendicular to the lines of action of the forces, then Fc A AC=L1; CB=L ̧; AB = L ̧ + LÂ; and consequently FA: FB: FC::CB: AC: AB; so that each of the three forces is proportional to the distance between the lines of action of the other two; and if any three parallel forces balance each other, they must be equivalent to two couples, as shown in the figure. 40. Resultant of Two Parallel Forces.-The resultant of any two of the three forces FA, FB, Fc, is equal and opposite to the third. Hence the resultant of two parallel forces is parallel to them, MOMENT OF A FORCE. 27 and in the same plane; if they act in the same direction, then their resultant is their sum, acts in the same direction, and lies between them; if they act in opposite directions, their resultant is their difference, acts in the direction of, and lies beyond, the preponderating force; and the distance between the lines of action of any two of those three forces-the resultant and its two components -is proportional to the third force. In order that two opposite parallel forces may have a single resultant, it is necessary that they should be unequal, the resultant being their difference. Should they be equal, they constitute a couple, which has no single resultant. 41. Resultant of a Couple and a Single Force in Parallel Planes.— Let M denote the moment of a couple applied to a body (fig. 9); and at a point O let a single force F be applied, in a plane parallel to that of the couple. For the given couple substitute an equivalent couple, consisting of a force - F equal and directly opposed to F at O, and a force F applied at A, the arm AO M being = , and of course par F A Fig. 9. 0 allel to the plane of the couple distance O A-to the left if M is right handed-to the right if M is left-handed. F X 42. Moment of a Force with respect to an Axis. -Let the straight line F represent a force applied to a body. Let OX be any straight line perpendicular in direction to the line of action of the force, and not intersecting it, and let A B be the common perpendicular of those two lines. At B conceive a pair of equal and directly opposed forces to be applied in a line of action parallel to F, viz.: FF, and -F=-F. The supposed application of such a pair of balanced forces does not alter the statical condition of the body. Then the original single force F, applied in a line tra Fig. 10. versing A, is equivalent to the force Fapplied in a line traversing B the point in OX which is nearest to A, combined with the couple composed of F and F', whose moment is F AB.. This is called the moment of the force F relatively to the axis OX, and sometimes also, the moment of the force F relatively to the plane which contains O X, and is parallel to the line of action of the force. If from the point B there be drawn two straight lines BD and BE, to the extremities of the line F representing the force, the area of the triangle BDE being = | F∙AB, represents one-half of the moment of F relatively to OX. 43. Equilibrium of any System of Parallel Forces in One Plane. -In order that any system of parallel forces whose lines of action are in one plane may balance each other, it is necessary and sufficient that the following conditions should be fulfilled : I. (As already stated in Art. 38) that the algebraical sum of the forces shall be nothing:- II. That the algebraical sum of the moments of the forces relatively to any axis perpendicular to the plane in which they act shall be nothing: two conditions which are expressed symbolically as follows:let F denote any one of the forces, considered as positive or negative, according to the direction in which it acts; let y be the perpendicular distance of the line of action of this force from an arbitrarily assumed axis O X, y also being considered as positive or negative, according to its direction; then, For, by the last Article, each force F is equivalent to an equal and parallel force F applied directly to OX, combined with a couple yF; and the system of forces F', and the system of couples y F, must each be in equilibrio, because when combined they are equivalent to the balanced system of forces F. In summing moments, right-handed couples are usually considered as positive, and left-handed couples as negative. 44. Resultant of any Number of Parallel Forces in One Plane.—The resultant of any number of parallel forces in one plane is a force in the same plane, whose magnitude is the algebraical sum of the magnitudes of the component forces, and whose position is such, that its moment relatively to any axis perpendicular to the plane in which it acts is the algebraical sum of the moments of the component forces. Hence let F, denote the resultant of any number of parallel forces in one plane, and y, the distance of the line of MOMENTS OF A FORCE. 29 action of that resultant from the assumed axis O X to which the positions of forces are referred: then = In some cases, the forces may have no single resultant, F being 0; and then, unless the forces balance each other com pletely, their resultant is a couple of the moment 2. y F. 45. Moments of a Force with respect to a Pair of Rectangular Axes -In fig. 11, let F be any single force; O an arbitrarily-assumed point,called the "originof co-ordinates;" - X 0 + X, − Y 0 + Y, a pair of axes traversing O, at right angles to each other and to the line of action of F. Let AB=y, be the common perpendicular of F and OX ; let AC = x, be the common perpendicular of F and OY. x and y are the "rectangular co-ordinates" of the line of action of F relatively to the axes - XO X, – YO + Y, respectively. According to the arrangement of the axes in the figure, x is to be considered as positive to the right, and negative to the left, of YO+ Y; and y is to be considered as positive to the left, and negative to the right, of - XO + X; right and left referring to the spectator's right and left hand. In the particular case represented, x and y are both positive. Forces, in the figure, are considered as positive upwards, and negative downwards ; and in the particular case represented, F is positive. -- Fig. 11. +x At B conceive a pair of equal and opposite forces, F' and - F', to be applied; F' being equal and parallel to F, and in the same direction. Then, as in Article 42, F is equivalent to the single force FF applied at B, combined with the couple constituted by F and - F with the arm y, whose moment is y F; being positive in the case represented, because the couple is right-handed. Next, at the origin Ò, conceive a pair of equal and opposite forces, F" and - F", to be applied, F" being equal and parallel to F and F', and in the same direction. Then the single force F is equivalent to the single force F" = F = F applied at O, combined with the couple constituted by F' and F" with the arm OB = x, whose moment is -F; being negative in the case represented, because the couple is left-handed. Hence it appears finally, that a force F acting in a line whose co-ordinates with respect to a pair of rectangular axes perpendicular to that line are x and y, is equivalent to an equal and parallel force acting through the origin, combined with two couples whose moments are, y F relatively to the axis OX, and x F relatively to the axis OY; right-handed couples being considered positive; and + Y lying to the left of +X, as viewed by a spectator looking from + X towards O, with his head in the direction of positive forces. 46. Equilibrium of any System of Parallel Forces.-In order that any system of parallel forces, whether in one plane or not, may balance each other, it is necessary and sufficient that the three following conditions should be fulfilled : I. (As already stated in Art. 38), that the algebraical sum of the forces shall be nothing: II. and III. That the algebraical sums of the moments of the forces, relatively to a pair of axes at right angles to each other, and to the lines of action of the forces, shall each be nothing :conditions which are expressed symbolically as follows:2. F= 0; y F = 0; 2x F = 0; for by the last Article, each force F is equivalent to an equal and parallel force F" applied directly to O, combined with two couples, y F with the axis OX, and x F with the axis OY; and the system of forces F", and the two systems of couples y F and x F, must each be in equilibrio, because when combined they are equivalent to the balanced system of forces F. 47. Resultant of any Number of Parallel Forces.-The resultant of any number of parallel forces, whether in one plane or not, is a force whose magnitude is the algebraical sum of the magnitudes of the component forces, and whose moments relatively to a pair of axes perpendicular to each other and to the lines of action of the forces, are respectively equal to the algebraical sums of the moments of the component forces relatively to the same axes. Hence let F, denote the resultant, and x, and y, the co-ordinates of its line of action, then In some cases, the forces may have no single resultant, · F |