A Treatise on the Theory of Screws

76 THE THEORY OF SCREWS. [85, the same as those of the original equation. The required theorem will therefore be proved if it can be shown that all the roots of this equation are real. That this is so is shown in Salmon’s Modern Higher Algebra, Lesson VI.* 86. Another investigation of the Principal Screws of Inertia. The n Principal Screws of Inertia can also be investigated in the following fundamental manner by the help of Lagrange’s equations of motion in generalized co-ordinates. Let £... be the co-ordinates (§ 95) of the impulsive screw. Let </>!,... fa, be the co-ordinates of the body, then fa,... fa, will be the co-ordinates of the instantaneous screw, and from Lagrange’s equations, d (dT\ dT dt\d<j>J dfa~ where T is the kinetic energy and where denotes the work done in a twist bfa against the wrench. If the screws of reference be co-reciprocal and if £" be the intensity of a wrench on £ then Pi^pdf'ti- As we are considering the action of only an impulsive wrench the effect of which is to generate a finite velocity in an infinitely small time we must have the acceleration infinitely great while the wrench is in action. The term is therefore negligible in comparison with and hence for the a(Pi at \dfa/ impulsive motion ■( d (dT\ dt \dfa) P1^ ’ We may regard £ and £" as both constant during the indefinitely small time e of operation of the impulsive wrench, whence (§ 79) 2^- ^///_ 1 dl Pi dfa Hence replacing fa,... <£„ by 3lt ... 3n we deduce the following (§§ 95, 96). If T be the kinetic energy of a body with freedom of the nth order, twisting about a screw 3 whose co-ordinates referred to any n co-reciprocals belonging to the system expressing the freedom are 3lt...3n, then the co-ordinates * See also Williamson and Tarleton’s Dynamics, 2nd edition, p. 457 (1889), and Routh’s Rigid Dynamics, Part II, p. 49 (1892). t Niven, Messenger of Math., May 1867, quoted by Routh, Rigid Dynamics, Part I, pp. 327-8.