A Treatise on the Theory of Screws

140 THE THEORY OF SCREWS. [150- In the limit we allow P and Q to coalesce, in which case, of course, P' and Q' coalesce, and p and q become coincident; but obviously we have then PQ : ML :: PX : LX, P'Q' : ML :: P’Y : LY; P'Q__P'Y LX PQ PX* LY’ P’ Yoc —and PX x , JU I JLjA. whence and as we have finally The result is, of course, the same as that of § 141. Being given the impulsive screw corresponding to P, we find P' by drawing PXL and LYP' ■, and then to produce a unit twist velocity on P', the intensity of the impul- sive wrench on P must be proportional to LX LY. It is obvious that by a proper adjustment of the units of length, force and twist velocity, LX may be the intensity of the impulsive wrench, and LY the acquired twist velocity. 151. Principal Screws of the Potential. Let us suppose that a body having two degrees of freedom is in a position of stable equilibrium under the influence of a conservative system of forces. If the body be displaced by a small twist, it will no longer be in a position of equilibrium, and a wrench has commenced to act upon it. This wrench can always, by suitable composition with the reactions of the constraints, be replaced by an equivalent wrench on a screw of the cylindroid (see § 96). For every point H, corresponding to a displacement screw, we have a related point, H’, corresponding to the screw about which the wrench is evoked. The relation is of the one-to-one type, and it will now be proved that the system of screws H is homographic with the corresponding system H'. The proof is obtained in the same manner as that already given in § 137, for impulsive and instantaneous screws. Let E be a displacement screw about which a twist of unit magnitude evokes a wrench of intensity e on E’; let f be the similar quantity for another pair of screws, F and F'. A twist of unit amplitude about H may be decomposed into components, HF HE EF’ EF’ about E and F, respectively.