274 DOC.
285
DECEMBER
1916
are
the
general
integral
of
(3)
and
a1...a2n
integration
constants.
I
set
ƒ
L
dt
=
Û(ax.
. .
0¡2n,
í),
(5)
or
if I
regard
the
aj's
from
(4) as
functions of
xk,
yk,
Ù(ai.
. .
a2n, t)
=
Q(xk,
yk,
t).
Similarly,
if
we
write
H(ai.
. .
a2n,
t)
=
H(xk,
ÿk,
t),
then
(with
the aid
of
(3)
and
(4))[2]
dH
_
ydH_dx^
dH_dyk_
_
y
_dy^dx^
dxk
dyk
daj
”
dxk
daj
dyk
daj
^
dt
daj
dt
ddj
d (x^~
®xk
\
d (
^
_
dxk
dai
V
*
*
dt
/
dt
VT
Vkdotj
is
the
result.
However,
_
dxk
^Vk~dt
-
H
=
L(xkyk, t)
-
dú
dt
Thus,
instead
of
(6)
one can
write
d
^
_
dxk
d2û
~ñ+
dk
o
-
dt
t”'
daj_
dajdt
and from
this
follows
dxk
dÙ
k
E
=
^
+
M°i
• •
•
«a») U
=
i,
• •
•
»)
If to these
n
equations
dxk
dú
Çs'k'âT
= âF
+
/ffj
is
added,
they
are
then
equivalent
to
the
equation
E
ÿkdxk
=
dÛ
+
E
Ajdaj
+
H
dt.
(6)
(7)