The composition and resolution of couples is easily shown to be done in a manner which perfectly resembles that of ROTATIONS. When the couples can have a common axis (act in the same plane or parallel planes), the moment of the resultant is, in sign and magnitude, the sum of the moments of the components, with their proper signs. To find the resultant of two couples which cannot have a common axis, take axes to them which pass through the same point, and on these axes lay down lines representing the moments of the couples in their proper direction. On those lines complete a parallelogram ; the direction of the diagonal is the axis of the resulting couple, and its length represents the moment of that couple. Care must be taken to lay down the directions of the moments properly on the axes ; the best isolated rule (when reference is not made to distinct co-ordinate plaues) is as follows : Let the parts of the plane of the axes which lie in the angle made by the lines representing moments be turned by the two couples in opposite directions. To the student to whom such a direction would be useful, we should say, appeal in all cases to the perceptions derived from ROTATION.
To apply the preceding theorems to the statics of a rigid body, we first take the following conventions :—Assume an origin and three rectangular axes of co-ordinates, as usual. Let the forces which act at each point of the system be decomposed into three, parallel to the axes of x, y, and z. Let each force be called positive, when it acts towards the positive part of the axis to .which it is parallel; if, for instance, the axis of z be vertical, and if its positive part tend upwards, all forces in the direction of z, wherever they act, are called positive while they act upwards, and negative when downwards. As to couples, let their moments be called positive when, acting in the planes of x and y, y and z, z and x, they tend to turn the positive part of the first named towards the positive part of the second (xy, ya, zw). Let r, be the first point of the system ; let its coordinates be y„ let the forces in the three directions acting at that point be x„ z,. Let be the second point; x„ y,, its co-ordinates ; x„ y„ ;, the forces there applied ; and so on. All co-ordinates and forces have their proper signs. At the origin apply the following pairs of equilibrating forces : x, and ; and —Y„, a, and —.z, ; x, and and z, and and so ou ; which of course do not affect the equilibrium, and are over and above those already applied. Again, at the extremity of x„ in the axis of x, apply the equilibrating forces ; at tho extremity of y„ in the axis of y, apply z„ —z, ; at the extremity of z„ in the axis of z, apply —x, ; and so on for the other points, Lastly, let the points of application of the original forces X,, Y„ z„ be changed so that each shall act at the projection of the point of application made by its co-ordinate ; and the same for the other points. Nothing is
done but the application of mutually destroying forces, or the change of the point of application of a force to another point in its direction, and the following figure will show the present arrangement for one point. The original forces, transferred, are marked x,Y,Z; the original point of application, 1'; and the other forces, equilibrating two and two, have great and small letters at their extremities.
We now see that the forces x, v, z, are equivalent to— I. Tho forces x, Y, z (marked a, B, c) applied at tho origin.
2. A pair of couples to the axis of z b) (x, n), the first positive with the moment YS, the second negative with the moment xy. These two are equivalent to one couple with the moment YX— Xy.
3. A pair of couples to the ells of a c) (r, 1), the total moment of which is ry— ris 4. A pair of couples to the axis of y (N, a) (z, et), the total moment of which is xi —z.r. Apply this to every point in the system, and let Ix stand for x, + x,+, &c., and so on : hence it appears that the whole of the forces are splivalent to forces Ix, Zr, 22, applied at the origin In the directions of x, y, and r, together with couples in the planes of 'y, y:, rsr, of which the momenta aro Then it appears that all the forces can be reduced to one force, v, acting at the origin, making angles with the axes whose cosines are A : v, B : v, C: r ; and one couple having a moment w, and whose axis makes with the axes of co-ordinates angles whose cosines are L w, :sv, x : w. But when there is equilibrium, both the force and the moment of the couple must vanish, for the single force cannot equal. brate a couple. Consequently the conditions of equilibrium are r = 0, = 0, which give A 0, n s 0, C = 0, L = 0, at = 0,B = 0, the six well-known conditions of equilibrium.
The forces will have a single resultant when v falls in the plane of the couple whose moment is w ; that is. when the direction of V is at right angles to the axis of the couple. This takes place when AL + BM + CB = 0, a well-known condition.
(For further information, we may refer to Poinsot's Elemens de Statigue ; or, in English, to Pratt's Mathematical Principles of Natural Philosophy ; or Pritchard's Theory of Couples.)