. EMPLOYMENT OF HIGHER AXOIDS. 165
difficulties, a circumstance which helps to explain the existing im- perfect comprehension of those even which have been hitherto made. The modern sewing-machine manufacture, and in part also that of agricultural machinery, have empirically and unknown to themselves made very satisfactory progress in the employment of the higher axoids ; the former with special skill, for it has already brought to considerable perfection the formation of complex enveloping surfaces.
The illustrations which we have used in the foregoing paragraphs have been in great part, although not entirely, drawn from the methods used for constructing wheel-teeth, and must be therefore more or less known, if not entirely familiar to those readers who have made machine -construction a subject of scientific study* The methods of procedure, however, deserve renewed attention, for they have now been developed in the special light of the general fundamental principles upon which they rest. The question has here been treated as one not of rules for constructing wheel-teeth, but of their general correspondence to a great principle. "We found that by a quite small extension of the ideas contained in them, methods are available generally which are commonly stated and understood as very limited rules. I trust therefore that such previous familiarity with particular instances will only have made it more easy to understand the general case and their relation to it.
I hope now to have made completely intelligible the fact that the construction of pairs of elements is possible for any motion, however complex, that is, that in all cases suitable profiles -can be determined for those elements. We have seen also that this problem may be solved in an immense variety of different ways, even in the case of the simpler motions and those more often occurring. While in former centuries the most distinguished geometrician occupied himself with separate solutions of detached problems, and necessarily regarded them as important propositions, he has to-day presented to him a limitless perspective, which appears almost more simple in its universality than the single case appeared before, and which affords rich opportunities to the practical mind in the determination of the best solution among the immense number of possible ones.
Perhaps I must fear that I have wearied my readers by these investigations, in which we have apparently progressed more point
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