DOC. 2 CO VARIANCE PROPERTIES
11
are
the transformation formulas.
J
and
J'
shall
be
the values of
the
integral
above
relative to K and
K',
respectively.
This
gives
Considering that
/1T
dr
is a
scalar, the transformation of
J'
in terms of the
coordinate system K gives
hence
In
further
calculations
we
shall
assume
that the coordinate
systems
K and K'
differ
only by infinitesimals,
i.e.,
the transformation is infinitesimal. We then have
to set
therefore
[p. 221]
and
where the
Axv
are understood
as
infinitesimal quantities whose squares and products
are negligible. This results in
Partial
integration
turn this into
(3)
[18]
We notice
that
the first
two
integrals can
be written
as
surface
integrals
which
dx'
dXP
~
__
__
`SrI
____
lv
2pr.J'=-
f
^
(*r
«
*.,»
"
ft
«
^
(Pm
«A
f
O
l»M
^
«
*»
?
»V
»))
'
f
V^^Yit
A
(PmrPn
?
9m
n)
A
(*"
/
A =
xv
-
dxv7
-xv
vX0\^Xy)
*P
=
Wfi
~
»"
dx'r ™"
°*p
dx^~'
md
Ä a
*
d(4xv)
xp*~dx"
"»z«"1"
dx"
'
where
the
A
ft/ "
"STL,
"
dVtnd*(4Xm)
J
/
,Yik9mn
rfiXu
()xtdXi
mnikt
Partial
J-if
mnikt
4/2M^9y«gm"8-£%JXm)dT 4/Ya9mJ-£r)4xm-d*'
mnikt
We
4/Ya9mJ-£r)4xm-d*'