Hence the locus of J relative to AB, and the locus relative to CD are equal ellipses of which A, B and C, D are respectively the foci. It may be noticed that the lamina in fig. 9 is not, strictly speaking, fixed, but admits of infinitesimal displacement, whenever the directions of the three links are concurrent (or parallel). [Illustration: FIG. 13.] The matter may of course be treated analytically, but we shall only require the formula for infinitely small displacements. If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are [lambda], [mu], and if the axes are simultaneously turned through an angle [epsilon], the co-ordinates of a point of the lamina, relative to the original axes, are changed from x, y to [lambda] + x cos [epsilon] - y sin [epsilon], [mu] + x sin [epsilon] + y cos [epsilon], or [lambda] + x - y[epsilon], [mu] + x[epsilon] + y, ultimately. Hence the component displacements are ultimately [delta]x = [lambda] - y[epsilon], [delta]y = [mu] + x[epsilon] (1) If we equate these to zero we get the co-ordinates of the instantaneous centre. § 4. _Plane Statics._--The statics of a rigid body rests on the following two assumptions:-- (i) A force may be supposed to be applied indifferently at any point in its line of action. In other words, a force is of the nature of a "bound" or "localized" vector; it is regarded as resident in a certain line, but has no special reference to any particular point of the line. (ii) Two forces in intersecting lines may be replaced by a force which is their geometric sum, acting through the intersection. The theory of parallel forces is included as a limiting case. For if O, A, B be any three points, and m, n any scalar quantities, we have in vectors m · [->OA] + n·[->OB] = (m + n) [->OC], (1) provided m · [->CA] + n·[->CB] = 0. (2) Hence if forces P, Q act in OA, OB, the resultant R will pass through C, provided m = P/OA, n = Q/OB; also R = P·OC/OA + Q·OC/OB, (3) and P·AC : Q·CB = OA : OB. (4) These formulae give a means of constructing the resultant by means of any transversal AB cutting the lines of action. If we now imagine the point O to recede to infinity, the forces P, Q and the resultant R are parallel, and we have R = P + Q, P·AC = Q·CB. (5) [Illustration: FIG. 14.] When P, Q have opposite signs the point C divides AB externally on the side of the greater force. The investigation fails when P + Q = 0, since it leads to an infinitely small resultant acting in an infinitely distant line. A combination of two equal, parallel, but oppositely directed forces cannot in fact be replaced by anything simpler, and must therefore be recognized as an independent entity in statics. It was called by L. Poinsot, who first systematically investigated its properties, a _couple_.