KINETIC ENERGY OF ELECTRICITY IN MOTION [I]
9
of the closing of the circuit and to the velocity of the needle.
If the former be 7, the latter v, and the logarithmic decrement
during closing X, the magnitude of such an impact is
-
λ
4 TV.
T
If a,, a, be the preceding and succeeding elongations the
needle reaches its position of rest with a velocity a,q/x and
leaves it with a velocity a/x. As the increase of velocity is
nearly uniform, we must put for v the mean value (a₁g+a,)/2x,
and thus the magnitude of the impact is
N'T
C
− 217 (a,q+a,) = − = (a,q+a,).
-
TK
K
By adding this increase of velocity to that caused by the
impact due to induction we obtain the equations
or
-
а₂ = h₁x+α₁q − c(a₂+a,q),
-
а₂ = h₂x+α₂q − c(a¸+a,q), etc.,
(1 + c)α₂ = xh₂+ (1 − c)a¸q,
(1+c)α¸ = xh₂+(1 − c)ᄶ, etc. ;
and by a similar calculation to that above
-
-
(1+c)a, − (1 − c )qa₁ = x(k₂+k₂),
-
(1+c)a, − (1 −c)qa, = x(k₂+kg),
-
(1 + c)a, − (1 − c )qan-1 = x(kn-1+kin).
If instead of the quantities k₁, k₂
we write their
theoretical value k, we get after a simple transformation
the equations
-
a₂−qa₁ +c(a₂+1α₁) = 2kx,
-
a¸ ¬ qa₂+c(a₂+qa,) = 2kx,
angan-1+c(an+qan-1) = 2kx;
and from these the most probable values of the unknown
quantities k, etc. must be calculated by the method of least
squares.