Next, consider any continuous motion of the lamina. The latter may be brought from any one of its positions to a neighbouring one by a rotation about the proper centre. The limiting position J of this centre, when the two positions are taken infinitely close to one another, is called the _instantaneous centre_. If P, P´ be consecutive positions of the same point, and [delta][theta] the corresponding angle of rotation, then ultimately PP´ is at right angles to JP and equal to JP·[delta][theta]. The instantaneous centre will have a certain locus in space, and a certain locus in the lamina. These two loci are called _pole-curves_ or _centrodes_, and are sometimes distinguished as the _space-centrode_ and the _body-centrode_, respectively. In the continuous motion in question the latter curve rolls without slipping on the former (M. Chasles). Consider in fact any series of successive positions 1, 2, 3... of the lamina (fig. 11); and let J12, J23, J34... be the positions in space of the centres of the rotations by which the lamina can be brought from the first position to the second, from the second to the third, and so on. Further, in the position 1, let J12, J´23, J´34 ... be the points of the lamina which have become the successive centres of rotation. The given series of positions will be assumed in succession if we imagine the lamina to rotate first about J12 until J´23 comes into coincidence with J23, then about J23 until J´34 comes into coincidence with J34, and so on. This is equivalent to imagining the polygon J12 J´23 J´34 ..., supposed fixed in the lamina, to roll on the polygon J12 J23 J34 ..., which is supposed fixed in space. By imagining the successive positions to be taken infinitely close to one another we derive the theorem stated. The particular case where both centrodes are circles is specially important in mechanism. [Illustration: FIG. 12.] The theory may be illustrated by the case of "three-bar motion." Let ABCD be any quadrilateral formed of jointed links. If, AB being held fixed, the quadrilateral be slightly deformed, it is obvious that the instantaneous centre J will be at the intersection of the straight lines AD, BC, since the displacements of the points D, C are necessarily at right angles to AD, BC, respectively. Hence these displacements are proportional to JD, JC, and therefore to DD´ CC´, where C´D´ is any line drawn parallel to CD, meeting BC, AD in C´, D´, respectively. The determination of the centrodes in three-bar motion is in general complicated, but in one case, that of the "crossed parallelogram" (fig. 13), they assume simple forms. We then have AB = DC and AD = BC, and from the symmetries of the figure it is plain that AJ + JB = CJ + JD = AD.