A Treatise on the Theory of Screws

220 the theory of screws. [213, We can also investigate the conditions under which five forces applied to a free rigid body can neutralize each other. The five forces must, as the body is free, belong to a screw system of the fourth order. Draw the cylindroid reciprocal to the system. The five forces must, therefore, intersect both the screws of zero pitch on the cylindroid. We thus prove the well-known theorem that if five forces equilibrate two straight lines can be drawn which intersect each of the five forces. Four of the forces will determine the two transversals, and therefore the fifth force may enjoy any liberty consistent with the requirement that it also intersects the same two lines. If Ai,...A5 be the five forces, the ratio of any pair, let us say for example, Ai : A2 is thus determined. Let P, Q be the two screws of zero pitch upon the cylindroid, i.e. the two common transversals of J.,,... As. Choose any two screws X and F reciprocal to both Aj and but not reciprocal to A3, At or As. Choose any screw Z reciprocal to Ait yl5) but not reciprocal to At or Construct (§ 25) the single screw I reciprocal to the five screws X, Y, P, Q, Z. The four screws X, Y, P, Q are reciprocal to the cylindroid Alt A2- therefore I, which is reciprocal to X, Y, P, Q, must lie upon the cylindroid (Ai, A2) (§ 24). Since P, Q, Z are all reciprocal to A3, A4, As, it follows that I being leciprocal to P, Q, Z must belong to the screw system A3, A4, As. Hence I is found on the cylindroid (Alt A2), and it must also belong to the system (-4s> Ait As). If, therefore, forces along ... ^ls are to equilibrate, the forces along Alt A2 must compound into a wrench on I. But I being known by construction the angles A.J and AJ are known and consequently the ratio of the sines of these angles, i.e. the relative intensities of the forces on A, and are determined (§ 14). If a free rigid body is acted upon by five forces, the preceding con- siderations will show in what manner the body could be moved so that it shall not do work against nor receive energy from any one of the forces. Let Ai,... As be the five forces. Draw two transversals L, M intersecting Ay ...A4. Construct the cylindroid of which L, M are the screws of zero pitch; find, upon this cylindroid, the screw X reciprocal to 45. Then the