A Treatise on the Theory of Screws

85] THE PRINCIPAL SCREWS OF INERTIA. 75 which is of course SpA</>! = 0; whence 3 and </, are reciprocal, and the same being true for each pair of principal screws of inertia we thus learn that they form a co-reciprocal system. We can also show that each pair of the Principal Screws of Inertia are Conjugate Screws of Inertia. It is easy to see that X pl^iPi i X y _Pl 'h'lpl____ (K—x'^p^—X x" — X’~'p1 — x" (pi—X^^pi — x'') As each of the terms on the left-hand side of this equation is zero, the expression on the right-hand is also zero, but this is equivalent to tpi'difa = 0; whence we show that 3 and <f> are conjugate screws of inertia and the required theorem has been proved. 85. An algebraical Lemma. Let U and V be two homogeneous functions of the second degree in n variables. If either U or V be of such a character that it could be expressed by linear transformation as the sum of n squares, then the discriminant of U + XK when equated to zero gives an equation of the «th degree in Å of which all the roots are real *. Suppose that V can by linear transformation assume the form a.1!2 + X2 ... + Xn!, and adopt xlt x2... xn as new variables, so that U — (IjlX^ 4" “I“ 26112^*1^'2 • • • • The discriminant of & + XF will, when equated to zero, give the equation for X, I ^ii ~I- ^12 > * * * ^in = o, tl2l 5 ^22 4" X., . . . Ct>2n I : • ; Ct>m ) ^n2 > * * • ^nn 4“ and the discriminant being an invariant the roots of this equation will be * A discussion is found in Zanchevsky, Theory of Screws and its application to Mechanics, Odessa 1889. Mr G. Chawner has most kindly translated the Russian for me.