80
DOC.
67 MARCH
1915
Moreover,
with
your expression
(78)
for
H[3]
and,
of
course, always
omitting
terms of
higher order,
d(Hy/g)
_
ldhfw
dg(r
]
~
2 dxa '
It
follows
(changing
g
to
-g)[4]
d(Hy/g)
v
d
(dHs/g\
^ dg^)
^
dxa
V
dfjp)
J
_2^2
(4)
where for
brevity
we
designate
with
A2
the
differential
operator
(Laplace’s)
*
d2
Z'7
1
dxV
Having
stated this
general theorem,
we
note
that
infinitesimal
transformation
(1)
will
be
adapted
if
the
y’s
satisfy
the
four
equations
(65a)[5]
s,
=
£
aav
d2
dxadxa
which in the
present
case
become
dHy/g\
dg(r}
)
d
dxv
=
0.
=
0,
(5)
Because
of
the
expressions
(3)
for
the
huv's,
these
are
linear
equations, homoge-
neous
and
of
the
fourth
order in
y.
They
are
therefore satisfied if for
the
y’s we
take
some
arbitrary polynomials
of
the
third
degree
in
x.
We
assume,
in
particular,
1
3
Vn
=
QCßXV
with
constant
cu
(not zero).
We
evidently
obtain
hfifJ CßX
^,
=
-2
c^;
therefore
^fifi =
d”C^
^
0,
q.e.d.
I
take the
opportunity
of
sending
you,
with
my compliments,
a
paper
of which
I
have
just
received
the
offprints.
With
warm
regards,
I
remain
yours sincerely,
T.
Levi-Civita.