A Treatise on the Theory of Screws

384 THE THEORY OF SCREWS. [349, on the cylindroid to which it is confined its specification merely gives a single datum. To be given a pair of corresponding Dynames is, however, to be given really two data—-one of these is for the screws themselves as before, while the other is derived from the ratio of the amplitudes. Thus while three pairs of corresponding screws amount to three data, two pairs of corresponding Dynames amount to no less than four data; the additional datum in this case enabling us to indicate the intensity of each correspondent as well as the screw on which it is situated. It can further be shown in the roost general case of the correspondence of the Dynames in two «-systems that the number of pairs of Dynames required to define the correspondence is one less than the number of pairs of screws which would be required to define merely a screw correspondence in the same two n-systems. In an n-system a screw has n — 1 disposable co-ordinates. To define the correspondence we require n +1 pairs of screws. Of course those on the first system may have been chosen arbitrarily, so that the number of data required for the correspondence is (n — 1) (n + 1) = n2 — 1. A Dyname in an «-system has n arbitrary data, viz., n — 1 for the screw, and one for the intensity: hence when we are given n pairs of corresponding Dynames we have altogether n2 data. We thus see that the n pairs of corresponding Dynames really contribute one more datum to the problem than do the n + 1 pairs of corresponding screws. The additional datum is applied in allotting the appropriate intensity to the sought Dyname. We can then use either the n pair of Dyname correspondents or the (n +1) pairs of screw correspondents. In previous articles we have used the latter; we shall now use the former. 350. Freedom of the fifth order. In the higher orders of freedom the screw correspondence does not indeed afford quite so simple a means of constructing the several pairs of corre- sponding screws as we obtain by the Dyname correspondence. In two five-systems the correspondence is complete when we are given five Dynames in one and the corresponding five Dynames in the other. To find the Dyname X in the second system, corresponding to any given Dyname A in the first system, we proceed as followsDecompose A into Dynames on the five screws which contain the five given Dynames on the first system. This is always possible, and the solution is unique. These components will correspond to determinate Dynames on the five corresponding screws: these Dynames compounded together will give the required Dyname X both in intensity and position. In the general case where a mass-chain possesses freedom of the fifth