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634           ON  THE EESULTANT  OF A NUMBER  OF  UNIT VIBRATIONS          [442
As a check upon (19) we may verify that it becomes unity when integrated with respect to as between — oo and + oo .    Starting from
and differentiating with respect to u, we get
- 4u5' f+c°
and differentiating again      —r /     as4e~ux*da>= 1. 6   6           STT^^-OO
Using these integrals in (19) with a = l, u~Qn, the required verification follows.
The above verification suggests a remark which may have a somewhat wide application. In many cases we can foresee that a facility function will have a form such as A eruyil dx, and then, since
it follows that A = ^(U/TT}. According to this law, the expectation of as is zero, but the expectation of a2 is finite. If we know this latter expectation, we may use the knowledge to determine u. For
Expectation of a? = 2 \/(u/ir)
We may take an example from the problem, just considered, of the position of the centre of gravity of points distributed along a line. If os-a ocz, ... xn be the coordinates of these points reckoned from the middle and CD that of the centre of gravity,
the integrations being in each case from — Ła to +%a.    Taking first the integration with respect to asn, we find that
a2 Mean x2 — -^— 2 + the corresponding expression with sc-n omitted,
so that                                   Mean xz = a2/12ri.
Accordingly u = Qnjo?, as in (19).
A similar argument might be employed for the law of facility of various resultants (r) of n unit vibrations with phases entirely arbitrary, starting with Ae~ur*rdr, and assuming 'that the mean value of r2 is n.
My principal aim in attacking the above problem was an introduction to the question of random vibrations when the phases of the unit components                a  V           2