A Treatise on the Theory of Screws

APPENDIX I. 489 If all the first minors following identity of the discriminant of S vanish we must have the -S «34 • •• a3n 0, s. ... 8, «43, ... adll «33, «34 ... a3n ®43> a44 • • ■ a4n ) ®n‘i> ^n4 • • • ®nn • • • ann by which we have S = ^33«32 + Aa2 + 2^34*3*4 • • • + ■ Hence U + XV = Assss2 + ^44s4a + 2yl34«a«4 ... + Anns,2. In the case of n = 3 we have U + XV = A33s2, which proves that V and U have double contact as we already proved in a different manner. In the general case all the differential coefficients of S will vanish if s3=0...sM=0, but these latter define a cyl indroid and therefore whenever the discriminant of s has two equal roots, every screw on a certain cylindroid is a principal Screw of Inertia. If the discriminant had three equal roots then S could be expressed in terms of s4, ...sn and in this case every screw on a certain 3-system would be a principal Screw of Inertia. If n — 1 of the roots of the discriminant were equal, then every (n — 2)nd minor would vanish, S would become the perfect square s,2 to a factor. And we have U+XV = A,msn2. In this case every screw of the n - 1 system defined by sn = 0 will be a principal Screw of Inertia. NOTE III. Twist velocity acquired by an impulsive wrench, § 90. The problem solved in § 90 may be thus stated. A body of mass M only free to twist about a is acted upon by a wrench of intensity r/" on a screw r;. Find the twist velocity acquired. From Lagrange’s equations we have, § 86