166 KINEMATICS OF MACHINERY.
by point than step by step, and have rather courted than shunned the difficulties of the problem. By degrees, however, we have completely proved the special laws upon which constrained pairs of elements can be constructed, and this was the end we had in view. These laws are not simple, not lying on the surface, but they are fixed and within the assumed limits universally applicable general laws. It may be well on this account to pause for a moment and to refer the reader once more to the nature of the ideas on this subject which have hitherto been held and which were sketched in the introduction, and especially to the somewhat extended system of Laboulaye of which I there promised further mention.
Let us ask ourselves what it is that Laboulaye's three systems levier, tour, and plan, which bear so great an apparent impress of geometric generalization - t really represent. We must look at this question in the light of the special acquaintance with the subject which it has been our object in the foregoing chapters to obtain.
In the first system the moving body has one fixed point, (le corps (a un point fixe), in the second two fixed points (le corps a deux points ou une droite fixe), in the third three fixed, or rather restrained points (I 'obstacle consiste en trois points fixes ou en un plan passant par ces trois points). We remark in the first place that Laboulaye bases his classification not upon kinematic chains, into which, as we have seen, the machine separates itself, but upon the pairs of elements themselves. For he speaks always of the restraint of a single body> not of one forming a part of a whole sytem of bodies. Let us then limit ourselves to this, although Laboulaye himself considers his system to represent machines generally. Which pair, however, has only one fixed point ? We have seen ( 5. IV;) that the fixing of one point only leaves the motion of the body to which it belongs quite indeterminate ; that neither a constrained chain nor a constrained pair of elements can be so formed. Laboulaye, however, chose as his illustration a swinging lever, one, that is, which turns about an axis. His system "levier " therefore coincides with the second of our lower pairs of elements, the turning-pair ( 15). But such a pair requires not one, but at least six points of restraint. It may of course be said that the fixed point in the systdme levier repre- sents a geometrical axis, and that Laboulaye's meaning, strictly
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