336 DOC.
71
PRINCETON LECTURES
THE GENERAL THEORY
We
have,
first,
by
(67),
AA*
=
-
(j)
TZßAadxß.
In
this,
TuaB
is
the value of
this
quantity at
the variable
point
G
of the
path
of
integration.
If
we
put
Eu =
Mo
-
Mp
and denote the value of
TuaB
at
P
by TuaB,
then
we
have,
with sufficient
accuracy,
TuaB
=
TuaB
+
aTuaB/axvEr.
Let, further, Aa
be
the value obtained
from
Aa
by
a
parallel displacement
along
the
curve
from
P
to G.
It
may
now
easily
be
proved
by means
of
(67)
that
Au
-
Au
is
infinitely
small
of the
first
order,
while,
for
a curve
of
infinitely
small
dimensions of
the first
order, AAu is
infinitely
small
of the
second
order. Therefore there
is
an error
of
only
the
second
order
if
we
put
[92]
a°
=
a*
-
r:rA*tT.
If
we
introduce
these
values of
TuaB
and
Aa
into the
integral,
we
obtain, neglecting
all
quantities
of
a
higher
order
than
the
second,
[93]
(85)
AAu
=
-
(aTuaB/sxa
-
TuaBTpaa)
A0
Q
EadaB.
The
quantity
removed
from
under the
sign
of
integration
refers
to
the
point
P.
Subtracting
1/2d(EaEß)
from
the
integrand,
we
obtain
i
j
(W
-
W).
[76]