supposed, where they can to some extent be used through their asymptotes. If for example the opposite links of the mechanism were made of equal length (Fig. 26), the centre lines of the four arms would form a parallelogram, and their intersections would always be at an infinite distance; so that both centroids become infinite and therefore cannot be drawn. The method of reduction which we have just used, if applied to this case, gives us two rolling circles of equal diameter. We can here, however, obtain the same result in a still clearer manner. For this purpose let equal circles i and k, Fig. 26, be described about a and /, with radii
less than half as long as a /, and let a straight line i k touch
them both externally. If now this line move without sliding upon the circles, they will turn about their centres a and / in the same way as they would if they were connected with the arms of the
Fig. 27.
parallelogram, that is, with a uniform angular velocity -ratio equal, in this case, to unity. Thus the three centroids, to which we have here reduced the original infinitely large curves, completely express the relative motion which it was desired to represent. Further, we are able to imagine them existing and moving simul- taneously with the original Curves. They second, as it were, the movement which has been set up, for which reason we may call them secondary centroids. It will be noticed that in this case we have not two but three connected figures, which is noteworthy; for we already know ( 8 above) that only a pair of (primary) centroids accompanies any relative motion of con-plane figures. That more than a pair of mutually rolling figures should result from the secondary representation of such a motion is not peculiar to this particular case, but is general.
