85] THE PRINCIPAL SCREWS OF INERTIA. 75
which is of course
SpA! = 0;
whence 3 and 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.