DIRECT AND INDIRECT SYNTHESIS. 529
directly with a pair of elements suited for any required plaee- or form-change. This is generally possible, for if we know the required motion we can determine (Chap. II.) its axoids, and these themselves may be used (as was shown in Chap. III.) as the profiles of the elements. In the cases, however, when the centroids are infinite ( 9) this cannot be done, and the problem requires a wider treatment, which presents the same difficulties as those of general direct synthesis. But it is not necessary to consider this further, for we saw long ago that the practical value of solutions of these problems by pairs of elements was far less than that of solutions based upon kinematic chains. This second method of synthesis, therefore, cannot furnish us with results of any practical value.
140. Indirect Kinematic Synthesis.
It is the province of the indirect synthetic method to give us beforehand the solutions of all those problems under which it is possible for the given problem to fall ; to solve, that is, all the problems of machine-kinematics in advance. At first sight this problem may seem so extended, in fact so measureless, that any attempt to solve it may appear to be nothing more than a mere theoretic proposition. "We must not forget, however, that some of the investigations we have already made point to the conclusion that the region covered by the problems of machine- kinematics is not unlimited. We may remember, for instance, the remarkable smallness of the number of lower pairs ( 15), or the definite number of mechanisms which can be formed from a given chain ( 3). So here also we shall find on closer examination that all these kinematic problems lie within a region which it is possible for us at least to survey. If the requirements of the case be not made too high, the difficulties attending the use of this form of synthesis, although great, are by no means insurmountable, espe- cially within the limits ordinarily covered by machine-construction.
That the special indirect synthesis is really practicable is proved by the results of our analytical investigations, Its function is to show us what kinematic pairs actually exist. Now we know
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