explains itself. Among all the kinematic chains just mentioned as those most generally and easily applied, the crank-chain is that which contains the pairs of elements,—the cylinder and the prism-pair, most suitable for chambering and for the making of fluid-tight joints. Invention has thus, unconsciously, fallen generally upon this chain.
We have seen at the same time how extremely important it is that the synthetic treatment should be carried out to the fullest extent possible, for it is full of promise of new and valuable results. The question is, what form this treatment should take; for what we have here been able to accomplish in this direction has brought us only to the outer limits of the subject. It might appear at first sight that the best plan would be to make "Synthetic Kinematics" a special subject of study and instruction, treating it in separate books, and working completely through it, pair by pair and chain by chain. I do not think, however, that this is the best method. It appears to me far more advisable that under "Applied Kinematics" we should treat mechanisms, which might then be arranged according to their practical applications, both analytically and synthetically. Synthesis should be here simply one of the aids in the investigation, not its governing idea; it must be used with and beside other methods, the whole being combined for the most advantageous treatment of each particular branch of the subject.
Another remark, however, must be made here. After the satisfactory consciousness which our investigation has given us that we are not working in a field of which we can never see the boundaries, there may arise a doubt whether the material now placed at our command may not too soon be exhausted, whether our scientific treatment of it may not speedily work the mine altogether out. The doubt is made all the stronger by the stress which we have laid upon the simplifications of the matter to which we have been able to make our way. It is not one, however, about which we need to trouble ourselves.
We have carried the synthesis far enough to allow us to look round, forwards and backwards, and to compare the ground which has been explored with that which still lies untouched before us. And in the latter we can see an immense, indeed, an inexhaustible series of problems awaiting the earnest investigator. The short sketch which we have given of the planet-wheel chains gives some
