DOC.
45
QUANTUM
THEOREM 443
where the
Rk
are
algebraic
functions of the
pi,
then
Zpidqi
is
always
a
complete
differential, provided
the
pi are
expressed
in
terms
of
the
qi
with the
help
of
(14).
{3}
The
quantum
condition demands that the
integral
fpi,dqi,
when extended
over an
irreducible
curve,
is
a multiple
of
h. This
quantum
condition coincides with the
one
by
Sommerfeld-Epstein
when, specifically,
each
pi
depends only upon
the
associated
qi.
If
there exist fewer than l
integrals
of
type
(14),
as
is the
case,
for
example,
according
to
POINCARE
in the
three-body problem,
then the
pi
are
not
expressible by
the
qi
and the
quantum
condition of
SOMMERFELD-EPSTEIN
fails also in the
slightly
generalized
form that has been
given
here.
Additional
notes
by
translator
{1}
"an"
and
"ßn"
have been corrected here
to
"al" and
"ßl."
{2}
The German word "rationell" is
an
economic
concept, quite
remote from
rational
numbers,
though
also derived from the Latin word "ratio." As will be
seen later,
Einstein has
a
geometric
rationalization in mind in order to facilitate
a more
concrete
and
descriptive understanding
of
an
otherwise abstract
entity
of
physics.
{3}
The index "i" in
"dqi"
was
missing.