A Treatise on the Theory of Screws

159] THE GEOMETRY OF THE CYLINDROID. 147 screw e not on $; for as e and any screw 7 on S are reciprocal to 01} 02, 03, 04, it will follow that any screw on the surface made from e and 7, just as S is made from a and ß, must also be reciprocal to 0lt 0.2, 03, 04. As 7 may be selected arbitrarily on 8, we should thus find that the screws reciprocal to öi, 0.2, 03, 04 were not limited to one surface, but constituted a whole group of surfaces, which is contrary to what has been already shown. It is therefore the same thing to say that a screw lies on 8, as to say that it is reciprocal to ^,^,^,^(§24). Since the condition of reciprocity involves the pitches of the two screws in an expression containing only their sum, it follows that if all the pitches on 04, 0.,, 03 , 04 be diminished by any constant m, and all those on S be increased by m, the reciprocity will be undisturbed. Hence, if the pitches of all the screws on S be increased by + m, the surface so modified will still retain the property, that twists about any three screws will neutralize each other if the amplitudes be properly chosen. We can now show that there cannot be more than two screws of equal pitch on 8; for suppose there were three screws of pitch m, apply the constant - m to all, thus producing on S three screws of zero pitch. It must therefore follow that three forces on S can be made to neutralize; but this is obviously impossible, unless these forces intersect in a point and lie on a plane. In this case the whole surface degrades to a plane, and the case is a special one devoid of interest for our present purpose. It will, however, be seen that in general S does possess two screws of any given pitch. We can easily show that a wrench can always be decomposed into two forces in such a way that the line of action of one of these forces is arbitrary. Suppose that S only possessed one screw X of pitch m. Reduce this pitch to zero; then any other wrench must be capable of decomposition into a force on X (i.e. a wrench of pitch zero), and a force on some other line which must lie on »S'; therefore in its transformed character there must be a second screw of zero pitch on »S', and, therefore, in its original form there must have been two screws of the given pitch m. Intersecting screws are reciprocal if they are rectangular, or if their pitches be equal and opposite ; hence it follows that a screw 0 reciprocal to 8 must intersect <8 in certain points, the screws through which are either at right angles to 0 or have an equal and opposite pitch thereto. From this we can readily show that S must be of a higher degree than the second ; for suppose it were a hyperboloid and that the screws lay on the generators of one species A, a screw 0 which intersected two screws of equal pitch m must, when it receives the pitch — m, be reciprocal to the entire system A. We can take for 0 one of the generators on the hyper- boloid belonging to the species B; 0 will then intersect every screw of the 10—2