A Treatise on the Theory of Screws

349] THE THEORY OF SCREW-CHÄINS. 383 obtain the following equations by simply adding the equations just written for each separate screw 2^ = (11) 20,+(12) + = (nl) 2Ö, + (w.2) S02... + (nn) 20}l. If the Bynames in the first system neutralize then their components on the screws of reference must vanish or 20,= 0, ... 20» = 0. But it is obvious from the equations just written that in this case S^ = 0,...2</>„ = 0, and therefore the corresponding Dynames will also neutralize. Given n pairs of corresponding Dynames in the two systems, we obtain n? linear equations which will be adequate to determine uniquely the n2 constants of the type (11), (12), &c. It is thus manifest that n given pairs of Dynames suffice to determine the Dyname in either system, corresponding to a given Dyname in the other. It is of course assumed that in this case the intensities of the two corresponding Dynames in each of the «-pairs are given as well as the screws on which they lie. 349. Properties of this correspondence. To illustrate the distinction between this Dyname correspondence and the screw correspondence previously discussed, let us take the case of two cylindroids. We have already seen that, given any three pairs of corre- sponding screws, the correspondence is then completely defined (§ 343). Any fourth screw on one of the cylindroids will have its correspondent on the other immediately pointed out by the equality of two anharmomc ratios. The case of the Dyname correspondence is, however, different inasmuch as we require more than two pairs of corresponding Dynames on the two cylindroids, in order to completely define the correspondence. For any third Dyname 0 on one of the cylindroids can be resolved into two Dynames and 0.2 on the two screws containing the given Dynames. These com- ponents will determine the components on the corresponding cylindroid, which being compounded, will give </> the Dyname corresponding to d. It is remarkable that two pairs of Dynames should establish the corre- spondence as completely as three pairs of screws. But it will be observed that to be given a pair of corresponding screws on the two cylindroids is in reality only to be given one datum. For one of the screws may be chosen arbitrarily; and as the other only requires one parameter to fix it