A Treatise on the Theory of Screws

81] THE PRINCIPAL SCREWS OF INERTIA. 71 80. Impulsive Screws and Instantaneous Screws. If a free quiescent rigid body receive an impulsive wrench on a screw ?/, the body will immediately commence to twist about an instantaneous screw a. The co-ordinates of a being given for the six screws of reference just defined, we now seek the coordinates of The impulsive wrench on ?; of intensity »/" is to be decomposed into com- ponents of intensities y'"ylt ... y'"ye on ay, ... a>0. The component on a>n will generate a twist velocity about <on amounting to 1 y"'yn M pn ’ but if d be the twist velocity about a which is finally produced, the expression just written must be equal to aan, and hence we have the following useful result:— If the co-ordinates of the instantaneous screw be proportional to alt ... a„, then the co-ordinates of the corresponding impulsive screw are proportional to ... pea6. 81. Conjugate Screws of Inertia. Let a be the instantaneous screw about which a quiescent body either free or constrained in any way will commence to twist in consequence of receiving an impulsive wrench on any screw whatever y. Let ß be the instantaneous screw in like manner related to another impulsive screw f. We have to prove that if £ be reciprocal to a then shall y be reciprocal to ß. When the body receives an impulsive wrench on f of intensity %1" there is generally a simultaneous reaction of the constraints, which takes the form of an impulsive wrench of intensity p" on a screw p. 1 he effect on the body is therefore the same as if the body had been free, but had received an impulsive wrench of which the component wrench on the first screw ot reference had the intensity ^"'^ + p'"pi- This and the similar quantities will be proportional to the co-ordinates of the impulsive screw which had the body been perfectly free would have ß as an instantaneous screw. These latter, as W6 have shown in §80, are proportional to pißi, pvßz ••• p^ßs* Hence it follows that, h being some quantity differing from zero, we have + p pi — hpißi, + p"Pi = hPiß» % "£<s + p p6 — hp^ßfi. Multiplying the first of these equations by p^, the second by^2«2, &c. adding