§ 28. Between the differentials of the original coordinates
x
a
{\displaystyle x_{a}}
x
a
′
{\displaystyle x'_{a}}
d
x
a
′
=
∑
(
b
)
π
b
a
d
x
b
{\displaystyle dx'_{a}=\sum (b)\pi _{ba}dx_{b}}
(30)
and formulae of the same form (comp. § 10) may be written down for the components of a vector expressed in
x
{\displaystyle x}
q
a
{\displaystyle \mathrm {q} _{a}}
−
g
′
=
p
−
g
{\displaystyle {\sqrt {-g'}}=p{\sqrt {-g}}}
we have according to (28)[ 1]
1
−
g
′
w
a
′
=
1
−
g
∑
(
b
)
π
b
a
w
b
{\displaystyle {\frac {1}{\sqrt {-g'}}}w'_{a}={\frac {1}{\sqrt {-g}}}\sum (b)\pi _{ba}w_{b}}
or
w
a
′
=
p
∑
(
b
)
π
b
a
w
b
{\displaystyle w'_{a}=p\sum (b)\pi _{ba}w_{b}}
Further we have for the infinitely small quantities
ξ
a
{\displaystyle \xi _{a}}
[ 2]
ξ
a
′
=
∑
(
b
)
p
b
a
ξ
b
{\displaystyle \xi '_{a}=\sum (b)p_{ba}\xi _{b}}
and in agreement with this for the components of a vector expressed in
ξ
{\displaystyle \xi }
Ξ
a
′
=
∑
(
b
)
p
b
a
Ξ
b
{\displaystyle \Xi '_{a}=\sum (b)p_{ba}\Xi _{b}}
so that we find from (25)[ 3]
χ
a
b
′
=
∑
(
c
d
)
p
c
a
p
d
b
χ
c
d
{\displaystyle \chi '_{ab}=\sum (cd)p_{ca}p_{db}\chi _{cd}}
Interchanging here
c
{\displaystyle c}
d
{\displaystyle d}
χ
a
b
′
=
∑
(
c
d
)
p
d
a
p
c
b
χ
d
c
=
−
∑
(
c
d
)
p
d
a
p
c
b
χ
c
d
{\displaystyle \chi '_{ab}=\sum (cd)p_{da}p_{cb}\chi _{dc}=-\sum (cd)p_{da}p_{cb}\chi _{cd}}
and
χ
a
b
′
=
1
2
∑
(
c
d
)
(
p
c
a
p
d
b
−
p
d
a
p
c
b
)
χ
c
d
{\displaystyle \chi '_{ab}={\frac {1}{2}}\sum (cd)\left(p_{ca}p_{db}-p_{da}p_{cb}\right)\chi _{cd}}
(31)
The quantity between brackets on the right hand side is a second order minor of the determinant
p
{\displaystyle p}
↑ Comp. § 7, l. c.
↑ For the infinitesimal quantities
x
a
{\displaystyle x_{a}}
x
a
′
=
∑
(
b
)
π
b
a
x
b
{\displaystyle x'_{a}=\sum (b)\pi _{ba}x_{b}}
and taking into consideration (19) and (20), i e.
ξ
a
=
∑
(
b
)
g
a
b
x
b
,
x
a
=
∑
(
b
)
γ
b
a
ξ
b
{\displaystyle \xi _{a}=\sum (b)g_{ab}x_{b},\ x_{a}=\sum (b)\gamma _{ba}\xi _{b}}
and formula (7) l. c, we may write (comp. note 2, p. 758, l. c.)
ξ
a
′
=
∑
(
b
)
g
a
b
′
x
b
′
=
∑
(
b
c
d
e
)
p
c
a
p
d
b
π
e
b
g
c
d
x
e
=
=
∑
(
c
d
)
p
c
a
g
c
d
x
d
=
∑
(
c
d
f
)
p
c
a
g
c
d
γ
f
d
ξ
f
=
∑
(
c
)
p
c
a
ξ
c
{\displaystyle {\begin{array}{l}\xi '_{a}=\sum (b)g'_{ab}x'_{b}=\sum (bcde)p_{ca}p_{db}\pi _{eb}g_{cd}x_{e}=\\\\\qquad =\sum (cd)p_{ca}g_{cd}x_{d}=\sum (cdf)p_{ca}g_{cd}\gamma _{fd}\xi _{f}=\sum (c)p_{ca}\xi _{c}\end{array}}}
↑ Put
Ξ
a
I
Ξ
b
I
I
=
ϑ
a
b
{\displaystyle \Xi _{a}^{I}\Xi _{b}^{II}=\vartheta _{ab}}
ϑ
a
b
′
=
Ξ
a
′
I
Ξ
b
′
I
I
=
∑
(
c
d
)
p
c
a
p
d
b
Ξ
c
I
Ξ
d
I
I
=
∑
(
c
d
)
p
c
a
p
d
b
ϑ
c
d
{\displaystyle \vartheta '_{ab}=\Xi _{a}^{'I}\Xi _{b}^{'II}=\sum (cd)p_{ca}p_{db}\Xi _{c}^{I}\Xi _{d}^{II}=\sum (cd)p_{ca}p_{db}\vartheta _{cd}}
and similar formulae for the other three parts of (25).
