A Treatise on the Theory of Screws

307] THE GEOMETRICAL THEORY. 331 the fourth order. It follows that, 3, cf>: a> must lie upon the same cylin- droid. Thus, 3, </> are the two screws required, and the problem has been solved. It is easily seen that there is only one such screw 3, and one such screw <f>. Or we might have proceeded as follows:—Take any three screws on P, and any three screws on Q. Then by a fundamental principle a wrench on o> can be decomposed into six component wrenches on these six screws. But the three component wrenches on P will compound into a single wrench on some screw 3 belonging to P. The three component wrenches on Q will compound into a single wrench on some screw </> belonging to Q. Thus the original wrench on &> may be completely replaced by single wrenches on 3 and (f>. But this proves that 3, <f>, and co are co-cylindroidal. In the special case of this theorem which we are now to use one of the systems of the third order assumes an extreme type. It consists simply of all possible screws of infinite pitch. The theorem just proved asserts that in this case a twist velocity about any screw a> can always be replaced by a twist velocity about some one screw belonging to any given system of the third order P, together with a suitable velocity of translation. In the problem before us we know three corresponding pairs of impulsive screws and instantaneous screws (r), a), (%, /3), (£, y), and we seek the impul- sive screw corresponding to some fourth instantaneous screw 8. It should be noticed that the data are sufficient but not redundant. We have seen how a knowledge of two pairs of corresponding impulsive screws and instantaneous screws provided eight of the coordinates of the rigid body. The additional pair of corresponding screws only bring one further co-ordinate. For, though the knowledge of 7 appropriate to a given f might seem five data, yet it must be remembered that the two pairs (?;, a) and (£ 7) must fulfil the two fundamental geometrical conditions, and so must also the two pairs (£, ß) and (f, y); thus, as 7 has to comply with four conditions, it really only conveys a single additional coordinate, which, added to the eight previously given, make the nine which are required for the rigid body. We should therefore expect that the knowledge of three corresponding pairs must suffice for the determination of every other pair. Let the unit twist velocity about 8 be resolved by the principles ex- plained in this section into a twist velocity on some screw belonging to a, ß, 7, and into a velocity of translation on a screw of infinite pitch. We have already seen that the impulsive screw corresponding to So must lie on the system of the third order defined by 77, and and that it is definitely determined. Let us denote by ß this known impulsive screw which would make the body commence to twist about 3„.