The sum of the moments of the two forces of a couple is the same about any point in the plane. Thus in the figure the sum of the moments about O is P·OA - P·OB or P·AB, which is independent of the position of O. This sum is called the _moment of the couple_; it must of course have the proper sign attributed to it. It easily follows that any two couples of the same moment are equivalent, and that any number of couples can be replaced by a single couple whose moment is the sum of their moments. Since a couple is for our purposes sufficiently represented by its moment, it has been proposed to substitute the name _torque_ (or twisting effort), as free from the suggestion of any special pair of forces. A system of forces represented completely by the sides of a plane polygon taken in order is equivalent to a couple whose moment is represented by twice the area of the polygon; this is proved by taking moments about any point. If the polygon intersects itself, care must be taken to attribute to the different parts of the area their proper signs. [Illustration: FIG. 21.] Again, any coplanar system of forces can be replaced by a single force R acting at any assigned point O, together with a couple G. The force R is the geometric sum of the given forces, and the moment (G) of the couple is equal to the sum of the moments of the given forces about O. The value of G will in general vary with the position of O, and will vanish when O lies on the line of action of the single resultant. [Illustration: FIG. 22.] The formal analytical reduction of a system of coplanar forces is as follows. Let (x1, y1), (x2, y2), ... be the rectangular co-ordinates of any points A1, A2, ... on the lines of action of the respective forces. The force at A1 may be replaced by its components X1, Y1, parallel to the co-ordinate axes; that at A2 by its components X2, Y2, and so on. Introducing at O two equal and opposite forces ±X1 in Ox, we see that X1 at A1 may be replaced by an equal and parallel force at O together with a couple -y1X1. Similarly the force Y1 at A1 may be replaced by a force Y1 at O together with a couple x1Y1. The forces X1, Y1, at O can thus be transferred to O provided we introduce a couple x1Y1 - y1X1. Treating the remaining forces in the same way we get a force X1 + X2 + ... or [Sigma](X) along Ox, a force Y1 + Y2 + ... or [Sigma](Y) along Oy, and a couple (x1Y1 - y1X1) + (x2Y2 - y2X2) + ... or [Sigma](xY - yX). The three conditions of equilibrium are therefore [Sigma](X) = 0, [Sigma](Y) = 0, [Sigma](xY - yX) = 0. (8) If O´ be a point whose co-ordinates are ([xi], [eta]), the moment of the couple when the forces are transferred to O´ as a new origin will be [Sigma]{(x - [xi]) Y - (y - [eta]) X}. This vanishes, i.e. the system reduces to a single resultant through O´, provided -[xi]·[Sigma](Y) + [eta]·[Sigma](X) + [Sigma](xY - yX) = 0. (9)