A Treatise on the Theory of Screws

232] FREEDOM OF THE FIFTH ORDER. 251 while they are also reciprocal to the cylindroid because they cross two screws thereon with pitches equal in magnitude but opposite in sign. They are therefore reciprocal to X. In like manner it can be shown that two of the other system of generators possess the same property. On every cylindroid there is as we know (§ 26) one screw of a given five- system. This important proposition may be otherwise proved as follows. Let 0 be the co-ordinates of a screw on the cylindroid, then these co-ordinates must satisfy four linear equations. There must be a fifth equation in the six quantities 0y, ... 0S inasmuch as 0 is to lie on the given five-system. Thus from these five equations one set of values of 0lt ... 06 can be determined. On a quadratic two-system (§ 224) there will always be two screws belonging to any given five-system. For the quadratic two-system is the surface whose screws satisfy four homogeneous equations of which three are linear and one is quadratic. If another linear equation be added two screws on the surface can, in general, be found which will satisfy that equation. 232. Impulsive Screws and Instantaneous Screws. We can determine the instantaneous screw corresponding to a given impulsive screw in the case of freedom of the fifth order by geometrical considerations. Let X, as before, represent the screw reciprocal to the freedom, and let p be the instantaneous screw which would correspond, to X as an impulsive screw, if the body were perfectly free; let y be the screw on which the body receives an impulsive wrench, and let £ be the screw about which the body would commence to twist in consequence of this impulse if it had been perfectly free. The body when limited to the screw system of the fifth order will commence to move as if it had been free, but had been acted upon by a certain unknown wrench on X, together with the given wrench on tj. The movement which the body actually acquires is a twisting motion about a screw 0 which must lie on the cylindroid (£, p). We therefore determine 0 to be that one screw on the known cylindroid (£, p) which is reciprocal to the given screw X. The twist velocity of the initial twisting motion about 0, as well as the intensity of the impulsive wrench on the screw X produced by the reaction of the constraints, are also determined by the same construction. For by § 17 the relative twist velocities about 0, %, and p are known; but since the impulsive intensity y"' is known, the twist velocity about £ is known (§ 90); and therefore, the twist velocity about 0 is known; finally, from the twist velocity about p, the impulsive intensity X'" is determined.