8
KINETIC ENERGY OF ELECTRICITY IN MOTION [I]
=
Further, let a₁, a, a, be successive elongations right and
left of the position of rest, a, α₁ + α₂, α₂ = α₂+a, etc., the
complete arcs of vibration, and k₁, k
the increments of
velocity in the position of rest, which measure the inductive
effects. Then, if for the present any special damping during
the impact be neglected
hence we get
=
а₂ = кk₂+qa₂ = «h₂+ «qk¸+q²α¸;
a₁ = xk₁+a₁(1+q),
a₂ = xk₂+xk₁₂(1+q) + a,q(1+g).
a2
If we multiply the first equation by q and subtract it from
the second, we find
and similarly
а₂ − α¸¶ = x(k₂+k₂),
а3 − a¸¶ = x(k₂+ lig),
an — an- 19 = x(kn-1 + kn).
Hence we find the mean value of the impacts k₁, k₂,
which should all be equal if the apparatus worked quite exactly
kk
=
+80+00
...
+ a₁-g(a + a + ... + an-1),
2(n-1)
or, if we denote the sum of all the complete arcs of vibration
by Σ,
kk =
-
(-a₁) — q(Σ - an)
2(n-1)
The application of this formula is very easy and is always
advisable when the separate impacts are not regular enough to
produce a constant limiting value of the arcs of swing, or
when for other reasons only a limited number of elongations
has been observed.
If in addition to the constantly occurring damping a
further damping occur during the instant of closing of the
circuit, this latter may be regarded as an impact in a direction
opposed to the motion which is proportional to the duration