96
CERTAIN IMPORTANT FUNCTIONS
etc., when is a continuous function of commencing with the value , or when we choose to attribute to a fictitious continuity commencing with the value zero, as described on page 90.
If we substitute in these equations the values of and which we have found, we get
|
(304)
|
|
(305)
|
where
may be substituted for
in the cases above described. If, therefore,
is known, and
as function of
,
and
may be found by quadratures.
It appears from these equations that is always a continuous increasing function of , commencing with the value , even when this is not the case with respect to and . The same is true of , when , or when if increases continuously with from the value .
The last equation may be derived from the preceding by differentiation with respect to . Successive differentiations give, if ,
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(306)
|
is therefore positive if
. It is an increasing function of
, if
. If
is not capable of being diminished without limit,
vanishes for the least possible value of
, if
. If
is even,
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(307)
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