A Treatise on the Theory of Screws

484 THE THEORY OF SCREWS. But since the seven screws are independent both and must be, in general, different from zero, whence by the former equation we have a!' ß" sin (ßp) sin (pa) ’ Thus we obtain the following theorem (§ 28). If seven wrenches on seven given screws equilibrate and if the intensity a" of one of the seven wrenches be given then the intensity of the wrench on any one ß of the remaining six screws can be determined as follows. Find the screw if/ reciprocal to the five screws remaining when a and ß are excluded from the seven. On the cylindroid (aß) find the screw p which is reciprocal to ip. Resolve the given wrench a" on a into component wrenches on ß and on p. Then the intensity of the component wrench thus found on ß is the required intensity ß" with its sign changed. NOTE II. Case of equal roots in the Equation determining Principal Screws of Inertia, § 86. We have already made use of the important theorem that if U and V are both homogeneous quadratic functions of n variables, then the discriminant of U + Å V when equated to zero must have n real roots for A provided that either U or V admits of being expressed as the sum of n squares (§ 85). The further important discovery has been made that whenever this deter- minantal equation has a repeated root, then every minor of the determinant vanishes (Routh, Rigid Dynamics, Part n. p. 51, 1892). This theorem is of much interest in connection with the Principal Screws of Inertia. The result given at the end of § 86 is a particular case. It may be further presented as follows. Taking the case of an n system each root of Å will give n equations -2Å01 = 1^, ...-2X0n = - Pl C^1 Pn d()n Of these n — 1 are in general independent and these suffice to indicate the values of 01; ... 0H. But in the case of a root once repeated the theorem above stated shows that we have not more than n - 2 independent equations in the series. The principal Screw of Inertia corresponding to this root is therefore indeterminate. But it has a locus found from the consideration that besides these n — 2 linear