By the combined effects of these contrary derangements of the extremities of the link from the vertical line, it might be expected that a point would exist, in the middle of the link, where the two contrary derangements would neutralise each other, and which point would therefore be expected to be disturbed neither to the right nor to the left, but to be moved upwards and downwards in the vertical line A B. Such is the principle of the parallel motion; and in fact the middle point of the link will move for all practical purposes accurately in the vertical line A B, provided that the angular play of the levers O P and _o p_ does not exceed a certain [Pg198] limit, within which, in practice, their motion may always be restrained. To trace the motion of the middle point of the link more minutely, let P P′ P″ P‴ be four positions of the lever O P, and let _p_ _p′_ _p″_ _p‴_ be the four corresponding positions of the lever _o p_. In the positions O P _o p_, the link will take the position P _p_, in which the entire link will be vertical, and its middle point _x_ will therefore be in the vertical line A B. When the one rod takes the position O P′, the other rod will have the position _o p′_; and the link will have the position P′ _p′_. The middle point of the link will be at _x′_, which will be found to be on the vertical line A B. Thus one half of the link P′ _x′_ will be to the left of the vertical line A B; while the other half, _p′ x′_, will be to the right of the vertical line; the derangement from the vertical line affecting each half of the link in contrary directions. Again, taking the one rod in the position O P″, the corresponding position of the other rod will be _o p″_, and the position of the link will be P″ _p″_. If the middle point of the link in this position be taken, it will be found to be at _x″_, on the vertical line A B; and, as before, one half of the link P″ _x″_ will be thrown to the left of the vertical line, while the other half _p″ x″_, will be thrown to the right of the vertical line. Finally, let the one rod be in its lowest position, O P‴, while the other rod shall take the corresponding position, _o p‴_. The direction of the link P‴ _p‴_ will now coincide with the vertical line; and its middle point _x‴_ will therefore be upon that line. The previous derangement of the extremities of the rod, to the right and to the left, are now redressed, and all the parts of the rod have assumed the vertical position. It is plain, therefore, that by such means the alternate motion of a point such as P or _p_, upwards and downwards in a circular arch, may be made to produce the alternate motions of another point _x_, upwards and downwards in a straight line.