A Treatise on the Theory of Screws

77 86] THE PRINCIPAL of an impulsive wrench by which the actual motion of the body could be produced are proportional to fdT___________ Pj ddf ■" pn ddn' 'Ehe existence of n Principal Screws of Inertia can now be readily deduced, for suppose that SCREWS OF INERTIA. 1 dT - 2X0, = -S , - 2X0» = fffi0 p1 d0j Pn aOn where X is an unknown factor. If then we make T = an0f + ai20f + 2a120,02... we have an equation of the nth degree for X as follows : du + pih,, ai2 ,... aln 0. d21 > ^22 + P^i • • • a2n Clnl , Cln2 > • • • ^nn 4" Pn\ It is essential to note that T is a function of such a character that by linear transformation it can be expressed as the sum of n squares, for suppose it could be expressed as Hf + Hf...-H,f, it would be possible to find a real screw which made If, H2, ... Hn-i each zero, and then the kinetic energy of the body twisting about that screw would be negative. Of course this is impossible. Hence we deduce from § 85 the important principle that all the Principal Screws of Inertia are real. If the equation had a repeated root the number of Principal Screws of Inertia is infinite. We take to = 4, but the argument applies to 3 and 2 also. (There can be no repeated root when n is either 5 or 6. See chaps. XVII. and XVIII.) We can choose variables such that T becomes M & (ufØf + uf Øf + uf Øf + uf Øf), and the pitch X becomes simultaneously pff+pßf.-.+pßf- If therefore the discriminant of T + Xp, equated to zero, has a pair of equal values for X, we must have a condition like P* ~ P* ‘ Take any screw of the system for which 03 = O, 04 = O, then T = M ØfufØf + ufØf), p = pff + p20f,