A Treatise on the Theory of Screws

APPENDIX I. 485 equations it must also satisfy the 6 - n linear equations which imply that it belongs to the w-system. In other words the co-ordinates satisfy (n - 2) + (6 — >i), i.e. four linear equations. But this is equivalent to saying that the screw lies on a cylindroid. We have thus the following result. In the case when the equation for \ has two equal roots, there must be n— 2 separate and distinct principal screws of inertia and also a cylindroid of which every screw is a principal Screw of Inertia. For every value of n from 1 to 6 it is of course known that the celebrated harmonic determinantal equation of § 86 has n real roots. But when the question arises as to the possibility of this equation as applied to our present problem having repeated roots, the several cases must be dis- criminated. It is to be understood that the body itself is to be of a general type without having e.g. two of the principal radii of gyration equal. The investigation relates to the possibility of a system of constraints which, while the body is still of the most general type, shall permit indeterminateness in the number of principal Screws of Inertia. Of course if n = 1 the equation is linear and has but a single root. n = 2. The equation under certain conditions may have two equal roots. n = 3. The equation under certain conditions may have two or three equal roots. n = 4. The equation under certain conditions may have two equal roots or three equal roots or four equal roots or two pairs of equal roots. n = 5. The equation can never have a i’epeated root. n = 6. The equation can never have a repeated root. Here comes in the restriction that the body is of a general type, for of course the last statement could not be true if two of the radii of gyration are equal or if one of them was zero. The curious contrast between the two last cases and those for the smaller values of n may be thus accounted for. The expression for T will contain terms and the ratios only being considered T will contain n(n+l) _ (n + 2)(n-l) distinct parameters. As a rigid body is specified both as to position and character by 9 co-ordinates, it follows that the coefficients of T are not unrestricted if (?i. + l) (h—1) greater t}lan 9. But this quantity is greater than 9 for the cases 2 of n = 5 and n = 6. We may put the matter in another way which will perhaps make it clearer. I shall take the two cases of n = 4 and n = 5.