Hence, since
[
Φ
2
.
β
]
×
[
Φ
.
β
]
{\displaystyle [\Phi ^{2}.\beta ]\times [\Phi .\beta ]}
[
Φ
.
β
]
×
β
{\displaystyle [\Phi .\beta ]\times \beta }
a
[
Φ
2
.
β
]
×
[
Φ
.
β
]
=
|
Φ
|
[
Φ
.
β
]
×
β
.
{\displaystyle a[\Phi ^{2}.\beta ]\times [\Phi .\beta ]=\left\vert \Phi \right\vert [\Phi .\beta ]\times \beta .}
Since
a
−
1
|
Φ
|
{\displaystyle a^{-1}\left\vert \Phi \right\vert }
p
2
=
a
−
1
|
Φ
|
.
{\displaystyle p^{2}=a^{-1}\left\vert \Phi \right\vert .}
If we also set
β
1
{\displaystyle \beta _{1}}
=
p
−
1
Φ
.
β
,
{\displaystyle =p^{-1}\Phi .\beta ,}
β
2
{\displaystyle \beta _{2}}
=
p
−
2
Φ
2
.
β
,
{\displaystyle =p^{-2}\Phi ^{2}.\beta ,}
etc.,
β
−
1
{\displaystyle \beta _{-1}}
=
p
Φ
−
1
.
β
,
{\displaystyle =p\Phi ^{-1}.\beta ,}
β
−
2
{\displaystyle \beta _{-2}}
=
p
2
Φ
−
2
.
β
,
{\displaystyle =p^{2}\Phi ^{-2}.\beta ,}
etc.,
the vectors
β
,
β
1
,
β
2
,
{\displaystyle \beta ,\beta _{1},\beta _{2},}
β
−
1
,
β
−
2
,
{\displaystyle \beta _{-1},\beta _{-2},}
α
′
,
{\displaystyle \alpha ',}
β
2
×
β
1
=
β
1
×
β
,
{\displaystyle \beta _{2}\times \beta _{1}=\beta _{1}\times \beta ,}
[
β
2
+
β
]
×
β
1
=
0.
{\displaystyle [\beta _{2}+\beta ]\times \beta _{1}=0.}
We may therefore set
β
2
+
β
=
2
n
β
1
.
{\displaystyle \beta _{2}+\beta =2n\beta _{1}.}
Multiplying by
p
−
1
Φ
,
{\displaystyle p^{-1}\Phi ,}
p
Φ
−
1
,
{\displaystyle p\Phi ^{-1},}
β
3
+
β
1
{\displaystyle \beta _{3}+\beta _{1}}
=
2
n
β
2
,
{\displaystyle =2n\beta _{2},}
β
4
+
β
2
{\displaystyle \beta _{4}+\beta _{2}}
2
n
β
3
,
{\displaystyle 2n\beta _{3},}
etc.,
β
1
+
β
−
1
{\displaystyle \beta _{1}+\beta _{-1}}
=
2
n
β
,
{\displaystyle =2n\beta ,}
β
+
β
−
2
{\displaystyle \beta +\beta _{-2}}
=
2
n
β
−
1
,
{\displaystyle =2n\beta _{-1},}
etc.,
Now, if
n
>
1
,
{\displaystyle n>1,}
β
,
β
1
,
β
2
{\displaystyle \beta ,\beta _{1},\beta _{2}}
β
−
1
,
β
−
2
{\displaystyle \beta _{-1},\beta _{-2}}
the broken line joining the termini of these vectors will be convex toward the origin. All these vectors must therefore lie between two limiting lines, which may be drawn from the origin, and which may be described as having the directions of
β
∞
{\displaystyle \beta _{\infty }}
β
−
∞
.
{\displaystyle \beta _{-\infty }.}
[1]
Φ
.
{\displaystyle \Phi .}
Φ
{\displaystyle \Phi }
n
<
−
1
{\displaystyle n<-1}
β
,
−
β
1
,
β
2
,
−
β
3
,
β
4
,
{\displaystyle \beta ,-\beta _{1},\beta _{2},-\beta _{3},\beta _{4},}
−
β
−
1
,
β
−
2
,
−
β
−
3
,
{\displaystyle -\beta _{-1},\beta _{-2},-\beta _{-3},}
except that a vector in the limiting directions will be reversed in direction by multiplication by
Φ
,
{\displaystyle \Phi ,}
If
1
>
n
>
−
1
,
{\displaystyle 1>n>-1,}
[2]
n
=
cos
q
.
{\displaystyle n=\cos q.}
Then
β
−
1
+
β
1
=
2
cos
q
β
.
{\displaystyle \beta _{-1}+\beta _{1}=2\cos q\,\beta .}
Let us now determine
γ
{\displaystyle \gamma }
β
1
=
cos
q
β
+
sin
q
γ
.
{\displaystyle \beta _{1}=\cos q\,\beta +\sin q\,\gamma .}
This gives
β
−
1
=
cos
q
β
−
sin
q
γ
.
{\displaystyle \beta _{-1}=\cos q\,\beta -\sin q\,\gamma .}
Now
α
′
{\displaystyle \alpha '}
α
,
β
,
{\displaystyle \alpha ,\beta ,}
γ
.
{\displaystyle \gamma .}
β
′
{\displaystyle \beta '}
γ
′
{\displaystyle \gamma '}
Ψ
=
cos
q
{
β
β
′
+
γ
γ
′
}
+
sin
q
{
γ
β
′
−
β
γ
′
}
,
{\displaystyle \Psi =\cos q\{\beta \beta '+\gamma \gamma '\}+\sin q\{\gamma \beta '-\beta \gamma '\},}
we have
Ψ
.
α
=
0
,
{\displaystyle \Psi .\alpha =0,}
Ψ
.
β
=
β
1
,
{\displaystyle \Psi .\beta =\beta _{1},}
Ψ
.
β
−
1
=
β
.
{\displaystyle \Psi .\beta _{-1}=\beta .}
↑ The termini of the vectors will in fact lie on a hyperbola.
↑ For the limiting cases, in which
n
=
1
,
{\displaystyle n=1,}
n
=
−
1
,
{\displaystyle n=-1,}
![{\displaystyle [\Phi ^{2}.\beta ]\times [\Phi .\beta ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/331e261879d622eee6f5836b9d42d927053ac24a)
![{\displaystyle [\Phi .\beta ]\times \beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00948ef3a24010aea83caf6bfab6f15bdf9472f)
![{\displaystyle a[\Phi ^{2}.\beta ]\times [\Phi .\beta ]=\left\vert \Phi \right\vert [\Phi .\beta ]\times \beta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43f0dc3d8762efc4fd90c2a634148d88cc2f22a1)
![{\displaystyle [\beta _{2}+\beta ]\times \beta _{1}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6050ce1d3fae1b13163c9bc6efe35502a6f001a5)
