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.