A Treatise on the Theory of Screws

490 THE THEORY OF SCREWS. But as the wrench is very great the initial acceleration, is great and conse- quently the second term on the left-hand side is negligible compared with the first. Whence ^-2^ajv"dt = 2^af", but P=Mua2å2(§ 89), whence 2 Mu? å = af", . 1 or • ria. nt a = M~^ • The kinetic energy = (§ 91> NOTE IV . Professor C. J. Jolys theory of the triple contact of conic and cubic. Professor C. J. Joly has pointed out to me that the conics of § 162 and § 165 which have triple contact with the nodal cubic are but particular instances of the more general theory which he investigates as follows. Let t be the parameter necessary to define a particular generator on a given cylindroid; we first show that the condition that a line, i.e. a screw 0 of zero pitch should intersect this generator may be expressed in the form at3 + bt? + ct + d = 0, where a, b, c, d are linear functions of the co-ordinates of 0 and, of course, functions also of the constants defining the cylindroid. For if a and ß be the two principal screws on the cylindroid then the co- ordinates of the screw e on the cylindroid making an angle A. with a are cos Åax + sin Xß1, ... cos Åa6 + sin Xßü, whence T^e0 = cos Å-srea + sin cos (<•$) = cos Å cos (ea) + sin Å cos (e/3), pe = cos2 Xpa + 2 cos X sin Xnraß + sin2 Xpß. If e and 0 intersect then 2-sree = cos(cö) pt, or putting t - tan Å, (cos (e/3)j»p-2OT-e/3) t3 + (cos (ea) pß + 2 COS (eß) — 2ffirea) t2 + (cos (eß) pa + 2 cos (ea) - 2-srej3) t + (cos (ea)pa — 2ra-ea) = 0, which has the form just given.