so that
−
4
π
w
h
=
k
1
−
ω
2
v
2
d
d
z
1
1
x
2
+
y
2
+
z
1
2
.
{\displaystyle -4\pi wh=k{\sqrt {1-{\frac {\omega ^{2}}{v^{2}}}}}{\frac {d}{dz_{1}}}{\frac {1}{\sqrt {x^{2}+y^{2}+z_{1}^{2}}}}.}
The displacement across any spherical surface must
=
e
{\displaystyle =e}
∫
(
x
f
+
y
g
+
z
h
)
d
S
=
a
e
;
{\displaystyle \int (xf+yg+zh)d\mathrm {S} =ae;}
and therefore, if
u
<
v
{\displaystyle u<v}
k
2
ω
∫
0
π
sin
θ
d
θ
(
1
−
u
2
v
2
sin
2
θ
)
3
2
=
e
(
1
−
u
2
v
2
)
3
2
;
{\displaystyle {\frac {k}{2\omega }}\int _{0}^{\pi }{\frac {\sin \theta \ d\theta }{\left(1-{\dfrac {u^{2}}{v^{2}}}\sin ^{2}\theta \right)^{\frac {3}{2}}}}={\frac {e}{\left(1-{\dfrac {u^{2}}{v^{2}}}\right)^{\frac {3}{2}}}};}
k
=
ω
e
(
1
−
u
2
v
2
)
−
1
2
.
{\displaystyle k=\omega e\left(1-{\frac {u^{2}}{v^{2}}}\right)^{-{\frac {1}{2}}}.}
Thus the lines of magnetic forces are circles round the axis of
z
{\displaystyle z}
ω
e
sin
θ
(
1
−
ω
2
v
2
)
r
2
{
1
−
ω
2
v
2
sin
2
θ
}
3
2
{\displaystyle {\frac {\omega e\sin \theta \left(1-{\dfrac {\omega ^{2}}{v^{2}}}\right)}{r^{2}\left\{1-{\dfrac {\omega ^{2}}{v^{2}}}\sin ^{2}\theta \right\}^{\frac {3}{2}}}}}
which is Mr. Heaviside's result. If
ω
>
v
{\displaystyle \omega >v}
sin
−
1
v
ω
=
β
{\displaystyle \sin ^{-1}{\dfrac {v}{\omega }}=\beta }
k
{\displaystyle k}
k
2
ω
∫
π
sin
−
1
ω
v
sin
θ
d
θ
(
1
−
u
2
v
2
sin
2
θ
)
3
2
=
e
(
1
−
u
2
v
2
)
3
2
;
{\displaystyle {\frac {k}{2\omega }}\int _{\pi }^{\sin ^{-1}{\frac {\omega }{v}}}{\frac {\sin \theta \ d\theta }{\left(1-{\dfrac {u^{2}}{v^{2}}}\sin ^{2}\theta \right)^{\frac {3}{2}}}}={\frac {e}{\left(1-{\dfrac {u^{2}}{v^{2}}}\right)^{\frac {3}{2}}}};}
or
k
=
ω
e
(
1
−
u
2
v
2
)
−
1
2
cos
β
(
1
−
ω
2
v
2
sin
2
β
)
1
2
.
{\displaystyle k={\frac {\omega e\left(1-{\dfrac {u^{2}}{v^{2}}}\right)^{-{\frac {1}{2}}}}{\cos \beta }}\left(1-{\frac {\omega ^{2}}{v^{2}}}\sin ^{2}\beta \right)^{\frac {1}{2}}.}
Thus the magnetic force
=
ω
e
(
1
−
u
2
v
2
)
1
2
(
1
−
ω
2
v
2
sin
2
β
)
1
2
cos
β
r
2
(
1
−
ω
2
v
2
sin
2
θ
)
3
2
{\displaystyle ={\frac {\omega e\left(1-{\dfrac {u^{2}}{v^{2}}}\right)^{\frac {1}{2}}\left(1-{\dfrac {\omega ^{2}}{v^{2}}}\sin ^{2}\beta \right)^{\frac {1}{2}}}{\cos \beta \ r^{2}\left(1-{\dfrac {\omega ^{2}}{v^{2}}}\sin ^{2}\theta \right)^{\frac {3}{2}}}}}
