A Treatise on the Theory of Screws

310] THE GEOMETRICAL THEORY. 335 about the several screws of the system of the third order, a geometrical construction could be obtained for determining the point corresponding to any instantaneous screw, when the point corresponding to the appropriate impulsive screw was known. We have now to introduce the simplification of the problem, which results when three of the principal screws of inertia of the body belong to the system. But a word of caution, against a possible misunderstanding, is first necessary. It is of course a fundamental principle, that when a rigid system has freedom of the «th order, there will always be, in the system of screws expressing that freedom, n screws such that an im- pulsive wrench administered on any one of those screws will immediately make the body begin to move by twisting about the same screw. These are the n principal screws of inertia. But in the case immediately under consideration the rigid body is sup- posed to be free, and it has, therefore, six principal screws of inertia. The system of the third order, at present before us, is one which contains three of these principal screws of inertia of the free body. Such a system of screws possesses the property, that an impulsive wrench on any screw belonging to it will set the body twisting about a screw which also belongs to the same system. This is the case even though, in the total absence of constraints, there is no kinematical difficulty about the body twisting about any screw whatever. As there are no constraints, we know that each instantaneous screw, of zero pitch, must be at right angles to the corresponding impulsive screw (§ 301). This condition will enable us to adjust the particular homography in the plane wherein each pair of correspondents represents an impulsive screw and the appropriate instantaneous screw. The conic, which is the locus of points corresponding to the screws of a given pitch p, has as its equation (§ 204) Pi0i + pftf + Pi6-i -P <A2 + = 0- The families of conics corresponding to the various values of p have a common self-conjugate triangle. The vertices of that triangle correspond to the three principal screws of inertia. The three points just found are the double points of the homography which correlate the points representing the impulsive screws with those representing the instantaneous screws. Let us take the two conics of the system, corresponding to p = 0 and p = 00 . They are Pi 0i + Pi Ø2 + Pi 0» = ° ........................................(i), 0? + 0/ + 0? = 0 .....................................__ (ii).