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J O dt C’
Again, because of the periodicity,
the first term on the right has the
same value at t = p and t = 0, and
hence only the second term remains on
the right side, Noting that
d d
- P (t - CT) = -- P(t - o)
dt &
and interchanging the order of inte-
gration, one transforms this term into
Since
and
K(C0) = 0
the term can also be written -
At this point, it is convenient,
though not essential, to choose the
1
PERIOD ENDING DECEMBER 10, 1952
units of P in such a way that log P
is always positive throughout the
oscillation. This is possible, since
P is bounded below. If log P is always
positive, a theorem(’) regarding
inequalities becomes applicable. It
has only to be noted that log P is a
monoton increasing function of P. The
theorem states that the expression
in the bracket is never negative. It
_____
.I__.. . .
is zero only if P(t - U ) = P(t) for
all t, that is, if (T is a multiple of
the period pa dK/do was assumed to
be nonpositive; hence, the whole
integral 8 is =< 0.
It has been shown before that, for
any periodic oscillation, the integral
over a period of the left side of
Eq. 7 is equal to zero. Hence, the
integral over the right side, which is
equal to the integral 8, has to vanish,
too. The condition for this is,
according to the above, that dK/do
vanish, except possibly at the points
_.I_____ -
(2)ti. H. Hardy, J. E. Littlewood. and ti. Pdlya,
Inequalatres, Zded., p. 278, Theorem 378, Cambridge
Univ. Press, 1952.
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