235] FREEDOM OF THE FIFTH ORDER. 253
We can easily verify as in § 84 that these five screws are co-reciprocal and
are also conjugate screws of inertia.
It is assumed in the deduction of this quintic that all the quantities
Xj-.-Xs are different from zero. If one of the quantities, suppose had
been zero this means that the first absolute principal Screw of Inertia
would belong to the ra-system expressing the freedom.
Let us suppose that = 0 then the equations are
Of course one solution of this system will be V" = 0, = 0 ... £6 = 0. This
means that the first absolute principal Screw of Inertia is also one of the
principal Screws of Inertia in the n-system, as should obviously be the case.
For the others £, = 0 and we have an equation of the fourth degree in
M22 + . + jW L 0
p2 - x pe-x
In the general case we can show that there are no imaginary roots in the
quintic, for since the screws Xi X<2 x6
Pi ~ ' P-2 - ’
Xi
Pi Pi — x'
are conjugate screws of inertia, we must have (§ 81)
If od = a + iß; y' — a — iß then this equation reduces to
X ____________=o
~(P1-ay + /32 ’
but as these are each positive terms their sum cannot be zero. This is a
particular case of § 86. (See Appendix, note 2.)
235. The limits of the roots.
We can now show the limits between which the five roots, just proved to
be real, must actually lie in the equation
, W 0 .
pj- ' pe — x ’