A Treatise on the Theory of Screws

376] THE THEORY OF PERMANENT SCREWS. 413 It has been shown in § 265 that if 0lt 02, ... 06 are the co-ordinates of a screw of zero pitch, then the co-ordinates of a screw of pitch px on the same axis are respectively a . Px dR a px dR a px dR “T — ■—r > t/2 H * • , • • • "I ~ • 4pj d0x " 4p2 d02 4p6 d06 In these expressions px denotes an arbitrary pitch, while R is the function 0* + 022... + 082 + 2 (12) ØÅ + 2 (23) ØÅ, &C„ where (12) is the cosine of the angle between the first and second screws of reference, and similarly for (23), &c. The principle just stated asserts that T must remain unchanged if we substitute for 0J 02, &c. the expressions 0 / + (0i + & e, 0' + (02 + e, &c. c 4P1 døj \ 4p2 d0.J We thus obtain the formula (å , P* dR\ dT /a p^dBxdT _ \ 4jPj døj dØi V 4epe døj dø,' As this must be true for every value of px, we must have, besides the vanishing emanant, the condition 1 dR dT , 1 dR dT _n px dØi dØi pe d06 dØs It is plain that this is equivalent to the statement that the. screw whose co-ordinates are 1 dR fdR l^dR Pi dØi P'a d02 Ps dØs must be reciprocal to the screw defined by the co-ordinates i £ dT 1 dT p2 dØi ’ p2 d02 ’ ’ ’ ’ ps dØs ' The former denotes a screw of infinite pitch parallel to 0 and hence it follows that the restraining screw must be perpendicular to 0. Remember- ing also that the restraining screw is reciprocal to 0, it follows that the restraining screw must intersect 0. We thus obtain the following result: If 0 be the screw about which a free rigid body is twisting, then to check the tendency of the body to depart from twisting about 0 a restraining wrench on a screw which intersects 0 at right angles must in general be applied.