LABOULAYE'S SYSTEM. 167
rendered, is that two points in this axis, which might be con- sidered as a kind of idealisation of the body, should be prevented from altering their position ; the two points coinciding of course in their normal projection. This cannot, however, be -the meaning, for it is the description Laboulaye himself gives of his second system, the systeme tour, he has therefore riot fallen into this error; he always speaks, too, of bodies, and not of their ideal representation by axes. But if any body is to be restrained we know that it must have a definite form, and that, being suitably formed, it must be restrained at at least six points. If we have one fixed point only the body must be spherical, it will require at least four points of restraint, and the motion which occurs is only so far constrained that the centre of the sphere cannot alter its position, and that the other points must move on spheric surfaces; with this limitation however they may have any possible motion.
To proceed : Laboulaye includes under systeme levier those pairs of elements which move by conic rolling, 11. That also is said distinctly. He does not however leave his chosen illustration the lever. " Le mouvement d'un point quelconque, appartenant au levier, sera de nature circulaire, en chaque instant et de plus en general alternatif dans une machine, se produisant le plus souvent dans un plan." We see that this definition fails altogether in clearness and certainty. Apparently it shadows forth in dim outline a pair of elements with a swinging motion, its appear- ance of deep and categoric generalisation has sometimes brought it into favour with mathematicians; it falls altogether to pieces, however, on a closer examination.
ISTor can the two other systems, tour and plan, fare better. In no one of the three systems is it made completely clear on the one hand what in strictness is meant by point fixe or plan inebranlable, or on the other hand what it is. that distinguishes absolutely one system from another. Let us reverse the question, and attempt to find in which of Laboulaye's classes one or other of our higher pairs of elements must be placed. We may choose the curve-triangle and square for instance, About this pair we know that if the square be fixed, then on motion taking place no point of the triangle remains in its original position. Every point in it moves. According to Laboulaye at least one point must remain stationary. The pair might perhaps be best placed under the
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