all the consequences which Chasles obtained artificially from the duality, and an immense number of others.
[I have used the wcrd train as synonymous with mechanism through this work. I regret that it was omitted, by an oversight, in line 8 of p. 47. I would venture to suggest, in connection with this subject, that it would be really more accurate, and in every way better, if the three-bar, five-bar, &c., linkworks and cells (about which so much that is interesting has recently been written and done by Peaucellier, Sylvester, Hart, Kempe, and others) should rather be called four-bar, six-bar, &c., linkworks respectively. The numbers hitherto attached to them are those of the moving links only of the kinematic chain. I need hardly point out that the plane in which the two fixed points are supposed to be, and relatively to which some other point describes some particular curve, is in every sense as much a "bar" as any of the other links. In view especially of this most interesting subject finding its way from the hands of the mathematicians to those of the students, it seems a pity that this most important fact should not be recognised in the names given to them. Prof. Sylvester distinguishes between the chain and the mechanism or train in the special cases mentioned by calling them linkage and linkwork respectively.]
9 (P. 72.) It seems suitable to call here particular attention to the fact that the two centroids are absolutely reciprocal, that is, that neither of them possesses, as a centroid, any property which the other has not. This may appear to be the case when (as in Fig. 21) one of the two curves is fixed. We see, however, from the above problem, that this difference of appearance may always be removed, and both the centroids made to move, if another link of the chain be fixed. The conditions of both pairs of centroids are identical, the fixing of one curve is merely accidental. The difference made by Poinsot between Poloid and Serpoloid is precisely the difference due to one of the curves being fixed,—it cannot, however, at least in the study of machines, be justified. It must be given up in other investigations also, I think, for there is no real difference between the two curves; the particular distinction made is indeed more apt to confuse than to explain, for it seems to point to the existence of only two centroids in a moveable system, while there is no such limit to their number; we have already noticed one case in which there were six pairs in one mechanism. Mechanisms such as those of Fig. 135 also—the spur-wheel train—give further illustrations of the undesirability of making this distinction. There the centroids of become a point, as do also those of ; they are therefore no longer visible as curves. The centroids of , on the other hand, the only ones remaining as curves in the chain, both move if the latter be placed on , and are therefore absolutely indistinguishable. The difference in name cannot therefore be logically justified, and in a Science, especially a young one, whatever cannot be logically justified should be carefully kept at a distance, and by no means taken up on mere grounds of convenience. I should not have spoken of the matter if it had not been for Prof. Aronhold's proposal to call the stationary and moving centroids by different names (Polbahn and Polkurve respectively), which has been too hastily accepted by the younger forces. [I believe Aronhold has now given up the use of the word Polkurve.] My experience has shown me a
