A Treatise on the Theory of Screws

BSSSaSaBSBHBHHHBHBHHBI 330 THE THEORY OF SCREWS. [306, ray in space, and assign to it an arbitrary pitch, the screw so formed may be regarded as an impulsive screw, and the corresponding instantaneous screw will not, in general, be defined. There is, however, a particular pitch for each such screw, which will constitute it a member of the system of the fifth order. It follows that any ray in space, when it receives the proper pitch, will be such that an impulsive wrench thereon would set any one of the singly infinite system of rigid bodies twisting about the same screw a. 307. The Geometrical Theory of Three Pairs of Screws. We can now show how, when three pairs of corresponding impulsive screws and instantaneous screws are given, the instantaneous screw, corre- sponding to any impulsive screw, is geometrically constructed. The solution depends upon the following proposition, which I have set down in its general form, though the application to be made of it is somewhat specialized. Given any two independent systems of screws of the third order, P and Q. Let co be any screw which does not belong either to P or to Q, then it is possible to find in one way, but only in one, a screw 3, belonging to P, and a screw (f>, belonging to Q, such that co, 3 and <f> shall all lie on the same cylin- droid. This is proved as follows. Draw the system of screws of the third order, P', which is reciprocal to P, and the system Q', which is reciprocal to Q. The screws belonging to P', and which are at the same time reciprocal to co, constitute a group reciprocal to four given screws. They, therefore, lie on a cylindroid which we call Po. In like manner, since Q is a system of the third order, the screws that can be selected from it, so as to be at the same time reciprocal to co, will also form a cylindroid which we call Q„. It is generally a possible and determinate problem to find, among the screws of a system of the third order, one screw which shall be reciprocal to every screw, on an arbitrary cylindroid. For, take three screws from the system reciprocal to the given system of the third order, and two screws on the given cylindroid. As a single screw can be found reciprocal to any five screws, the screw reciprocal to the five just mentioned will be the screw now desired. We apply this principle to obtain the screw 3, in the system P, which is reciprocal to the cylindroid Qo. In like manner, we find the screw cf>, in the system Q, which is reciprocal to the cylindroid P(l. From the construction it is evident that the three screws 3, $>, and co are all reciprocal to the two cylindroids Po and Qo. This is, of course, equivalent to the statement that 3, </>, co are all reciprocal to the screws of a system of