A Treatise on the Theory of Screws

416 THE THEORY OF SCREWS. [378- These equations enable us to study the correspondence between each instantaneous screw 3 and the corresponding restraining screw v- It is to be noted that this correspondence is not of the homographic, or one-to-one type, such as we meet with in the study of the Principal Screws of Inertia, and in other parts of the Theory of Screws. The correspondence now to be considered has a different character. 379. Limitation to the position of the Restraining Screw. If a particular screw 3 be given, then no doubt, a corresponding screw r/ is given definitely, but the converse is not true. If y be selected arbitrarily there will not in general be any possible 3. If, however, there be any one 3, then every screw on the same axis as 3 will also correspond to the same y. From the equations in the last article we can eliminate the six quantities, 3lt... 3ti; we can also write 17/' = y"y1, ■ ■ ■ Vn = v"Vn where y" is the intensity of the restraining wrench and ylt...ye the co-ordinates of the screw on which it acts. We have whence « W + Vi") = 2a6p2&>3 - 2acp3w2, a (ift" - %") = (b2 - c2) w2&>3, b2 - c2 Vi + y» _j)p2_cPs a Vl ~ Vi ®2 "s ’ and from the two similar equations we obtain, by addition, b2-c2 Vi + V2 + c2 - a2 y3 + m + a2 -J)2 Vn + y« _ q « Vi — yi b ys-yi c y5-ye It might at first have been supposed that any screw might be the possible residence of a restraining wrench, provided the corresponding instantaneous screw were fitly chosen. It should however be remembered that to each restraining screw corresponds a singly infinite number of possible instan- taneous screws. As the choice of an instantaneous screw has five degrees of infinity, it was to be presumed that the restraining screws could only have four degrees of infinity, i.e. that the co-ordinates of a restraining screw must satisfy some equation, or, in other words, that they must belong to a screw system of the fifth order, as we have now shown them to do. 380. A verification. We confirm the expression for the co-ordinates of y in the following manner. It has been shown (§376) that so long as 3 retains the same direction and situation, its pitch is immaterial so far as y is concerned. This might have been inferred from the consideration that a rigid body twisting about a screw has no tendency to depart from the screw in so far as its velocity of translation is concerned. It is the rotation which necessitates the restrain- ing wrench if the motion is to be preserved about the same instantaneous