Chapter XIX: Part 19

are equivalent to
_ _
/-l n[pi]x /-l n[pi]x
- | [f](x) cos ------ dx, + | [f](x) sin ------ dx,
_/0 l _/ 0 l

thus the series is
_ _ _
1 / l 1 [oo] n[pi]x / l n[pi]x 1 [oo] n[pi]x / l n[pi]x
-- | [f](x)dx + -- [Sigma] cos ------ | f(x) cos ------ dx + -- [Sigma] sin ------ | [f](x) sin ------ dx,
2l_/-l l 1 l _/-l l l 1 l _/-l l

which may be written
_ _
1 / l 1 [oo] / l n[pi](x - x')
-- | [f](x') dx' + -- [Sigma] | [f](x') cos ------------- dx'. (3)
2l_/-l l 1 _/-l l

The series (3), which represents a function [f](x) arbitrarily given
for the interval -l to l, is what is known as Fourier's Series; the
expressions (1) and (2) being regarded as the particular forms which
(3) takes in the two cases, in which [f](-x) = -[f](x), or [f](-x) =
f(x) respectively. The expression (3) does not represent f(x) at
points beyond the interval -l to l, unless [f](x) has a period 2l. For
a value of x within the interval, at which [f](x) is discontinuous,
the sum of the series may cease to represent [f](x), but, as will be
seen hereafter, has the value 1/2 {[f](x + 0) + [f](x - 0)}, the mean
of the limits at the points on the right and the left. The series
represents the function at x=o, unless the function is there
discontinuous, in which case the series is 1/2 {[f](+0) + [f](-0)};
the series does not necessarily represent the function at the points l
and -l, unless [f](l) = [f](-l). Its sum at either of these points is
1/2 {[f](l) + [f](-l)}.

_Examples of Fourier's Series._--(a) Let [f](x) be given from 0 to l,
by [f](x)=c, when 0 <= x < 1/2 l, and by f(x)= -c from 1/2 l to l; it
is required to find a sine series, and also a cosine series, which
shall represent the function in the interval.

We have
_ _ _
/ l n[pi]x /1/2 l n[pi]x / l n[pi]x
| [f](x) sin ------ dx = c | sin ------ dx - c | sin ------ dx
_/ 0 l _/ 0 l _/1/2 l l

cl
= ----- (cos n[pi] - 2 cos 1/2 n[pi] + 1).
n[pi]

This vanishes if n is odd, and if n = 4m, but if n = 4m + 2 it is
equal to 4cl/n[pi]; the series is therefore

4c /l 2[pi]x 1 6[pi]x 1 10[pi]x \
---- ( -- sin ------ + -- sin ------ + -- sin ------- + ... ).
[pi] \2 l 3 l 5 l /

For unrestricted values of x, this series represents the ordinates of
the series of straight lines in fig. 1, except that it vanishes at the
points 0, (1/2)l, l, (3/2)l ...

We find similarly that the same function is represented by the series

4c / [pi]x 1 3[pi]x 1 5[pi]x \
---- ( cos ----- - -- cos ------ + -- cos ------ - + ... )
[pi] \ l 3 l 5 l /

during the interval 0 to l; for general values of x the series
represents the ordinate of the broken line in fig. 2, except that it
vanishes at the points (1/2)l, (3/2)l....

(b) Let [f](x) = x from 0 to 1/2 l, and f(x) = l - x, from 1/2 l to l;
then
_ _ _
/ l n[pi]x / 1/2 l n[pi]x / l n[pi]x
| [f](x)sin ------ dx= | x sin ------ dx + | (l - x)sin ------ dx
_/ 0 l _/ 0 l _/1/2 l l

l^2 n[pi] l^2 n[pi] l^2n / n[pi] \
= - ------ cos ----- + --------- sin ----- + ----- (cos ----- - cos n[pi] )
2n[pi] 2 n^2[pi]^2 2 n[pi] \ 2 /

l^2 l^2 n[pi] l^2 n[pi] 2l^2 n[pi]
+ ----- cos n[pi] - ------ cos ----- + --------- sin ----- = --------- sin -----
n[pi] 2n[pi] 2 n^2[pi]^2 2 n^2[pi]^2 2

hence the sine series is

4l / nx 1 3[pi]x 1 5[pi]x \
------ (sin -- - --- sin ------ + --- sin ------ - ... )
[pi]^2 \ l 3^2 l 5^2 l /

For general values of x, the series represents the ordinates of the
row of broken lines in fig. 3.

The cosine series, which represents the same function for the interval
0 to l, may be found to be

1 2l / 2[pi]x 1 6[pi]x 1 10[pi]x \
-- l - ------ (cos ------ + --- cos ------ + --- cos ------- + ... )
4 [pi]^2 \ l 3^2 l 5^2 l /

This series represents for general values of x the ordinate of the set
of broken lines in fig. 4.

_Dirichlet's Integral._--The method indicated by Fourier, but first
carried out rigorously by Dirichlet, of proving that, with certain
restrictions as to the nature of the function [f](x), that function is
in general represented by the series (3), consists in finding the sum
of n+1 terms of that series, and then investigating the limiting value
of the sum, when n is increased indefinitely. It thus appears that the
series is convergent, and that the value towards which its sum
converges is 1/2 {[f](x + 0) + [f](x - 0)}, which is in general equal
to [f](x). It will be convenient throughout to take -[pi] to [pi] as
the given interval; any interval -l to l may be reduced to this by
changing x into lx/[pi], and thus there is no loss of generality.

We find by an elementary process that

1/2 + cos (x - x') + cos 2(x - x') + ... + cos n(x - x')

2n + 1
sin ------ (x' - x)
2
= -------------------.
2 sin 1/2(x' - x)

Hence, with the new notation, the sum of the first n+1 terms of (3) is
_
1 / [pi] sin (2n + 1)/2 (x' - x)
---- | [f](x') ----------------------- dx'.
[pi]_/-[pi] 2 sin 1/2 (x' - x)

If we suppose [f](x) to be continued beyond the interval -[pi] to
[pi], in such a way that [f](x) = [f](x + 2[pi]), we may replace the
limits in this integral by x + [pi], x-[pi] respectively; if we then
put x' - x = 2z, and let [f](x') = [F](z), the expression becomes
1/[pi] [int][-[pi]/2 to [pi]/2] F(z) (sin mz/sin z) dz, where m = 2n +
1; this expression may be written in the form
_ _
1 /[pi]/2 sin mz 1 /[pi]/2 sin mz
---- | F(z) ------ dz + ---- | F(-z) ------ dz. (4)
[pi]_/ 0 sin z [pi]_/ 0 sin z

We require therefore to find the limiting value, when m is
indefinitely increased, of [int][0 to [pi]/2] F(z)(sin mz/sin z) dz;
the form of the second integral being essentially the same. This
integral, or rather the slightly more general one [int][0 to h]
F(z)(sin mz/sin z) dz, when 0 < h <= 1/2[pi], is known as Dirichlet's
integral. If we write X(z)= F(z)(z/sin z), the integral becomes
[int][0 to h] X(z)(sin mz/z) dz, which is the form in which the
integral is frequently considered.

_The Second Mean-Value Theorem._--The limiting value of Dirichlet's
integral may be conveniently investigated by means of a theorem in the
integral calculus known as the second mean-value theorem. Let a, b be
two fixed finite numbers such that a<b, and suppose [f](x), [phi](x)
are two functions which have finite and determinate values everywhere
in the interval except for a finite number of points; suppose further
that the functions [f](x), [phi](x) are integrable throughout the
interval, and that as x increases from a to b the function [f](x) is
monotone, i.e. either never diminishes or never increases; the theorem
is that
_ _ _
/ b /[xi] / b
| [f](x) [phi](x) dx = [f](a + 0) | [phi](x) dx + [f](b - 0) | [phi](x) dx
_/ a _/ a _/[xi]

when [xi] is some point between a and b, and [f](a), [f](b) may be
written for [f](a + 0), [f](b - 0) unless a or b is a point of
discontinuity of the function [f](x).

To prove this theorem, we observe that, since the product of two
integrable functions is an integrable function, [int][a to b] [f](x)
[phi](x) dx exists, and may be regarded as the limit of the sum of a
series [f](x0) [phi](x0) (x1 - x0) + [f](x1) [phi](x1) (x2 - x1) + ...
+ [f]x(n-1) [phi]x_(n-1) (x_n - x_(n-1)) where x0 = a, x_n = b and x1,
x2 ... x_(n-1) are n - 1 intermediate points. We can express
[phi](x_r) (x_(r+1) - x_r) in the form Y_(r+1) - Y_r, by putting

K=r
Y_r = [Sigma] [phi](x_(K-1)) (x_K - x_(K-1)), Y0 = 0.
K=1

Writing X_r for [f](x_r), the series becomes

X0(Y1 - Y0) + X1(Y2 - Y1) + ... + X_(n-1)(Y_n - Y_(n-1))

or Y1(X0 - X1) + Y2(X1 - X2) + ... + Y_n(X_(n-1) - X_n) + Y_n X_n.

Now, by supposition, all the numbers Y1, Y2 ... Y_n are finite, and
all the numbers X_(r-1) - X_r are of the same sign, hence by a known
algebraical theorem the series is equal to M(X0 - X_n) + Y_n X_n,
where M is a number intermediate between the greatest and the least of
the numbers Y1, Y2, ... Y_n. This remains true however many partial
intervals are taken, and therefore, when their number is increased
indefinitely, and their breadths are diminished indefinitely according
to any law, we have
_ _
/ b _ / b
| [f](x)[phi](x)dx = {[f](a) - [f](b)} M + [f](b) | [phi](x) dx
_/ a _/ a

when M is intermediate between the greatest and least values which
[int][a to x] [phi](x) dx can have, when x is in the given integral.
Now this integral is a continuous function of its upper limit x, and
therefore there is a value of x in the interval, for which it takes
any particular value between the greatest and least values that it
has. There is therefore a value [xi] between a and b, such that

_
_ /[xi]
M = | [phi](x)dx,
_/ a

hence
_ _ _
/ b /[xi] / b
| [f](x) [phi](x) dx = {[f](a) - [f](b)} | [phi](x) dx + [f](b) | [phi](x) dx
_/ a _/ a _/ a
_ _
/[xi] / b
= [f](a) | [phi](x) dx + [f](b) | [phi](x) dx.
_/ a _/[xi]

If the interval contains any finite numbers of points of discontinuity
of [f](x) or [phi](x), the method of proof still holds good, provided
these points are avoided in making the subdivisions; in particular if
either of the ends be a point of discontinuity of [f](x), we write
[f](a + 0) or [f](b - 0), for [f](a) or [f](b), it being assumed that
these limits exist.

_Functions, with Limited Variation._--The condition that [f](x), in
the mean-value theorem, either never increases or never diminishes as
x increases from a to b, places a restriction upon the applications of
the theorem. We can, however, show that a function [f](x) which is
finite and continuous between a and b, except for a finite number of
ordinary discontinuities, and which only changes from increasing to
diminishing or vice versa, a finite number of times, as x increases
from a to b, may be expressed as the difference of two functions
[f]1(x), [f]2(x), neither of which ever diminishes as x passes from a
to b, and that these functions are finite and continuous, except that
one or both of them are discontinuous at the points where the given
function is discontinuous. Let [alpha], [beta] be two consecutive
points at which [f](x) is discontinuous, consider any point x1, such
that [alpha] <= x1 <= [beta], and suppose that at the points M1, M2
... M_r between [alpha] and x1, [f](x) is a maximum, and at m1, m2 ...
m_r, it is a minimum; we will suppose, for example, that the ascending
order of values is [alpha], M1, m1, M2, m2 ... M_r, m_r, x1; it will
make no essential difference in the argument if m1 comes before M1, or
if M_r immediately precedes x1, M_(r-1) being then the last minimum.

Let [psi](x1) = [[f](M1) - [f]([alpha] + 0)] + [[f](M2) - [f](m1)]+ ...
+[[f](M_r) - [f](m_(r-1))] + [[f](x1) - [f](m_r)];

now let (x1) increase until it reaches the value (M_(r+1)) at which
[f](x) is again a maximum, then let

[psi](x1) = [[f](M1) - [f]([alpha] + 0)] + [[f](M2) - [f](m1)] + ...
+ [[f](M_r) - [f](m_(r-1)] + [[f](M_(r+1)) - [f](m_r)];

and suppose as x increases beyond the value M_(r+1), [psi](x1) remains
constant until the next minimum m_(r+1) is reached, when it again
becomes variable; we see that [psi](x1) is essentially positive and
never diminishes as x increases.

Let

[chi](x1) = [[f](M1) - f(m1)] + [[f](M2) - [f](m1)] + ... + [[f](M_r)
- [f](m_r)],

then let x1 increase until it is beyond the next maximum M_(r+1), and
then let

[chi](x1) = [[f](M1) - [f](m1)] + [[f](M2) - [f](m1)] + ... +
[[f](M_r) - [f](m_r)] + [[f](M_(r+1)) - [f](x1)]

thus [chi](x1) never diminishes, and is alternately constant and
variable. We see that [psi](x1) - [chi](x1) is continuous as x1
increases from [alpha] to [beta], and that [psi](x1) - [chi](x1) =
[f](x1) - [f]([alpha] + 0), and when x1 reaches [beta], we have
[psi]([beta]) - [chi](x1) = [f]([beta] - 0) - [f]([alpha] + 0). Hence
it is seen that between [alpha] and [beta], [f](x) = [[psi](x) +
[f]([alpha] + 0)] - [chi](x), where [psi](x) + [f]([alpha] + 0),
[chi](x) are continuous and never diminish as x increases; the same
reasoning applies to every continuous portion of [f](x), for which
the functions [psi](x), [chi](x) are formed in the same manner; we now
take [f]1(x)=[psi](x) + [f]([alpha] + 0) + C, [f]2(x) = [chi](x) + C,
where C is constant between consecutive discontinuities, but may have
different values in the next interval between discontinuities; the C
can be so chosen that neither [f]1(x) nor [f]2(x) diminishes as x
increases through a value for which [f](x) is discontinuous. We thus
see that [f](x) = [f]1(x) - [f]2(x), where [f]1(x), [f]2(x) never
diminish as x increases from a to b, and are discontinuous only where
[f](x) is so. The function [f](x) is a particular case of a class of
functions defined and discussed by Jordan, under the name "functions
with limited variation" (_fonctions a variation bornee_); in general
such functions have not necessarily only a finite number of maxima and
minima.

_Proof of the Convergence of Fourier's Series._--It will now be
assumed that a function [f](x) arbitrarily given between the values
-[pi] and +[pi], has the following properties:--

(a) The function is everywhere numerically less than some fixed
positive number, and continuous except for a finite number of values
of the variable, for which it may be ordinarily discontinuous.

(b) The function only changes from increasing to diminishing or vice
versa, a finite number of times within the interval; this is usually
expressed by saying that the number of maxima and minima is finite.

These limitations on the nature of the function are known as
Dirichlet's conditions; it follows from them that the function is
integrable throughout the interval.

On these assumptions, we can investigate the limiting value of
Dirichlet's integral; it will be necessary to consider only the case
of a function F(z) which does not diminish as z increases from 0 to
1/2[pi], since it has been shown that in the general case the
difference of two such functions may be taken. The following lemmas
will be required:

1. Since
_ _
/[pi]/2 sin mz /[pi]/2 [pi]
| ------ dz = | {1 + 2cos 2z + 2cos 4z + ... + 2cos 2nz} dz = ----;
_/ 0 sin z _/ 0 2

this result holds however large the odd integer m may be.

[pi]
2. If 0 < [alpha] < [beta] <= ----,
2
_ _ _
/[beta] sin mz 1 /[gamma] 1 /[beta]
| ------ dz = ----------- | sin mz dz + ---------- | sin mz dz
_/[alpha] sin z sin [alpha]_/[alpha] sin [beta]_/[gamma]

where [alpha] < [gamma] < [beta], hence
_
| /[beta] sin mz | 2 / 1 1 \ 4
| | ------ dz | < -- ( ---------- + --------- ) < -------------;
| _/[alpha] sin z | m \sin[alpha] sin[beta]/ m sin [alpha]
_
| / [beta] sin mz | 4
a precisely similar proof shows that | | ------ dz | < --------,
| _/ [alpha] z | m[alpha]
_ _
/[beta] sin mz /[beta] sin mz
hence the integrals | ------ dz, | ------ dz, converge to
_/[alpha] sin z _/[alpha] z

the limit zero, as m is indefinitely increased.
_
| / [oo] sin [theta] |
3. If [alpha] > 0, | | ----------- d[theta] | cannot exceed
| _/[alpha] [theta] |

1/2[pi]. For by the mean-value theorem
_
| / h sin[theta] | 2 2
| | ---------- d[theta] | < ------- + --,
| _/[alpha] [theta] | [alpha] h
_
| / h sin[theta] | 2
hence | Lh = [oo] | ---------- d[theta] | <= -------;
| _/[alpha] [theta] | [alpha]
_
| /[oo] sin[theta] | 2 [pi]
in particular if [alpha] >= [pi] | | ---------- d[theta] | <= ---- < ----.
| _/[alpha] [theta] | [pi] 2
_
d /[oo] sin[theta] sin[alpha]
Again -------- | ---------- d[theta] = - ----------, [alpha] > 0,
d[alpha]_/[alpha] [theta] [alpha]
_
/[oo] sin[theta]
therefore | ---------- d[theta] increases as [alpha] diminishes,
_/[alpha] [theta]

when [theta] < [alpha] < [pi]; but lim
_ _
/[oo] sin[theta] [pi] | /[oo] sin[theta] | [pi]
| ---------- d[theta] = ----, hence | | ---------- d[theta] | < ----,
[alpha]=0_/[alpha] [theta] 2 | _/[alpha] [theta] | 2

where [alpha] < [pi], and < [pi]/2 where [alpha] >= [pi]. It follows that
_
| /[beta] sin[theta] |
| | ---------- d[theta] | <= [pi], provided 0 <= [alpha] < [beta].
| _/[alpha] [theta] |
_
/[pi]/2 sin mz
To find the limit of | F(z) ------ dz, we observe that it may be
_/ 0 sin z

written in the form
_ _
/[pi]/2 sin mz / [mu] sin mz
F(0) | ------ dz + | {F(z) - F(0)} ------ dz
_/ sin z _/ 0 sin z
_
/[pi]/2 sin mz
+ | {F(z) - F(0)} ------ dz
_/[mu] sin z

where [mu] is a fixed number as small as we please; hence if we use
lemma (1), and apply the second mean-value theorem,
_
/[pi]/2 sin mz [pi]
| F(z) ------ dz - ---- F(0)
_/ 0 sin z 2
_
/[mu] z sin mz
= | {F(z) - F(0)} ----- ------ dz
_/ [0] sin z z
_ _
/[xi]^1 sin mz /[pi]/2 sin mz
+ {F([mu] + 0) - F(0)} | ------ dz + {F (1/2[pi] - 0) - F(0)} | ------ dz
_/ [mu] sin z _/[xi]^1 sin z

when [xi]^1 lies between [mu] and 1/2[pi]. When m is indefinitely
increased, the two last integrals have the limit zero in virtue of
lemma (2). To evaluate the first integral on the right-hand side, let
G/z = {F(z) - F(0)} (sin z/z), and observe that G(z) increases as z
increases from 0 to [mu], hence if we apply the mean value theorem
_ _
| /[mu] sin mz | | /[mu] sin mz |
| | G([mu]) ------ dz| = |G([mu]) | ------ dz|
| _/ 0 z | | _/[xi] z |
_
| /m[mu] sin[theta] |
= |G([mu]) | ---------- d[theta]| < [pi] G([mu]),
| _/m[xi] [theta] |

where 0 < [xi] < [mu], since G(z) has the limit zero when z = 0. If
[epsilon] be an arbitrarily chosen positive number, a fixed value of
[mu] may be so chosen that [pi]G([mu)] < 1/2[epsilon], and thus that
_
| /[mu] sin mz |
| | G(z) ------ dz| < 1/2[epsilon].
| _/0 z |

When [mu] has been so fixed, m may now be so chosen that
_
| /1/2[pi] sin mz [pi] |
| | F(z) ------ dz - ---- F(0)| < [epsilon].
| _/0 sin z 2 |

It has now been shown that when m is indefinitely increased
_
/[pi]/2 sin mz [pi]
| F(z) ------ dz - ---- F(0) has the limit zero.
_/ 0 sin z 2

Returning to the form (4), we now see that the limiting value of
_ _
1 /[pi]/2 sin mz 1 /[pi]/2 sin mz
---- | F(z) ------ dz + ---- | F(-z) ------ dz
[pi]_/ 0 sin z [pi]_/ 0 sin z

1/2{F(+0) + F(-0)}; hence the sum of n + 1 terms of the series
_ _
1 / l 1 / l n[pi](x - x^1)
-- | [f](x) dx + -- [Sigma] | [f](x^1) cos ------------- dx
2l _/-l l _/-l l

converges to the value 1/2 {[f](x + 0) + [f](x - 0)}, or to [f](x) at
a point where [f](x) is continuous, provided [f](x) satisfies
Dirichlet's conditions for the interval from -l to l.

_Proof that Fourier's Series is in General Uniformly Convergent._--To
prove that Fourier's Series converges uniformly to its sum for all
values of x, provided that the immediate neighbourhoods of the points
of discontinuity of [f](x) are excluded, we have
_
| /[pi]/2 sin mz [pi] | 4
| | F(z)------ dz - ---- F(0)| < [pi]G ([mu]) + ---------- {F([mu] + 0) - F(0)}
| _/ sin z 2 | m sin [mu]

4
+ ------------ {F(1/2[pi] - 0) - F(0)}
m sin [xi]^1

[pi][mu] 4
< -------- {[f](x + 2[mu]) - [f](x)} + ---------- {[f](x + 2[mu]) - [f](x)}
sin [mu] m sin [mu]

4
+ ------------ {[f](x + [pi]) - [f](x)}
m sin [xi]^1

Using this inequality and the corresponding one for F(-z), we have

|S_(2n+1)(x) - [f](x)| < [mu] cosec [mu] [|[f](x + 2[mu]) - [f](x)|
+ |[f](x - 2[mu]) - [f](x)|] + A|m cosec [mu],

where A is some fixed number independent of m. In any interval (a, b)
in which [f](x) is continuous, a value [mu]1 of [mu] can be chosen
such that, for every value of x in (a, b), |[f](x + 2[mu]) - [f](x)|,
|[f](x - 2[mu]) - [f](x)| are less than an arbitrarily prescribed
positive number [epsilon], provided [mu] = [mu]1. Also a value [mu]2
of [mu] can be so chosen that [epsilon][mu]2 cosec [mu]2 < 1/2[eta],
where [eta] is an arbitrarily assigned positive number. Take for [mu]
the lesser of the numbers [mu]1, [mu]2, then |S_(2n+1) - [f](x)| <
[eta] + A|m cosec [mu] for every value of x in (a, b). It follows
that, since [eta] and m are independent of x, |S_(2n+1) - [f](x)| <
2[epsilon], provided n is greater than some fixed value n1 dependent
only on [epsilon]. Therefore S_(2n+1) converges to [f](x) uniformly in
the interval (a, b).

_Case of a Function with Infinities._--The limitation that [f](x) must
be numerically less than a fixed positive number throughout the
interval may, under a certain restriction, be removed. Suppose F(z) is
indefinitely great in the neighbourhood of the point z = c, and is

such that the limits of the two integrals [int][c to c[+-][epsilon]]
F(z) dz are both zero, as [epsilon] is indefinitely diminished, then

_
/[pi]/2 sin mz
| F(z) ------ dz
_/ 0 sin z

denotes the limit when [epsilon] = 0, [epsilon]^1 = 0 of
_ _
/c-[epsilon] sin mz /[pi]/2 sin mz
| F(z) ------ dx + | F(z) ------ dz,
_/ 0 sin z _/c+[epsilon]^1 sin z

both these limits existing; the first of these integrals has
1/2[pi]F(+0) for its limiting value when m is indefinitely increased,
and the second has zero for its limit. The theorem therefore holds if
F(z) has an infinity up to which it is absolutely integrable; this
will, for example, be the case if F(z) near the point C is of the form
x(z)(z - c)^-[mu] + [psi](z), where [chi](c), [psi](c) are finite, and
0 < [mu] < 1. It is thus seen that [f](x) may have a finite number of
infinities within the given interval, provided the function is
integrable through any one of these points; the function is in that
case still representable by Fourier's Series.

_The Ultimate Values of the Coefficients in Fourier's Series._--If
[f](x) is everywhere finite within the given interval -[pi] to +[pi],
it can be shown that a_n, b_n, the coefficients of cos nx, sin nx in
the series which represent the function, are such that na_n, nb_n,
however great n is, are each less than a fixed finite quantity. For
writing [f](x) = [f]1(x) - [f]2(x), we have
_ _ _
/[pi] /[xi] /[pi]
| [f]1(x) cos nxdx = [f]1(-[pi] + 0) | cos nxdx + [f]1([pi] - 0) | cos nxdx
_/-[pi] _/-[pi] _/[xi]

hence
_
/[pi] sin n[xi] sin n[xi]
| [f]1(x) cos nxdx = [f]1(-[pi] + 0) --------- + [f]1([pi] - 0) ---------
_/-[pi] n n

with a similar expression, with [f]2(x) for [f]1(x), [xi] being
between [pi] and -[pi]; the result then follows at once, and is
obtained similarly for the other coefficient.

If [f](x) is infinite at x = c, and is of the form [phi](x)/(x - c)^K
near the point c, where 0 < K < 1, the integral
_
/[pi]
| [f](x)cos nxdx contains portions of the form
_/-[pi]
_ _
/[epsilon]+[epsilon] [phi](x) / c [phi](x)
| --------- cos nxdx | --------- cos nxdx;
_/ [c] (x - c)^K _/c-[epsilon] (x - c)^K

consider the first of these, and put x = c + u, it thus becomes
_
/[epsilon] [phi](c + u)
| ------------ cos n(c + u) du, which is of the form
_/ 0 u^K
_
/[epsilon] cos n(c + u)
[phi](c + [theta][epsilon]) | ------------ du;
_/ 0 u^K

now let nu = v, the integral becomes
_ _ _ _
| cos nc /n[epsilon] cos v sin nc /n[epsilon] sin v |
[phi](c + [theta][epsilon]) | ------- | ----- dv - ------- | ----- dv |;
|_ n^(1-K) _/ 0 v^K n^(1-K) _/ 0 v^K _|

hence n^(1-K) [int]([pi] to -[pi]) [f](x) cos nxdx becomes, as n is
definitely increased, of the form
_ _ _ _
| /[oo] cos v /[oo] sin v |
[phi](c) | cos nc | ----- dv - sin nc | ----- dv |
|_ _/ 0 v^K _/ 0 v^K _|

which is finite, both the integrals being convergent and of known
value. The other integral has a similar property, and we infer that
n^(1-K) a_n, n^(1-K) b_n are less than fixed finite numbers.

_The Differentiation of Fourier's Series._--If we assume that the
differential coefficient of a function [f](x) represented by a
Fourier's Series exists, that function [f]'(x) is not necessarily
representable by the series obtained by differentiating the terms of
the Fourier's Series, such derived series being in fact not
necessarily convergent. Stokes has obtained general formulae for
finding the series which represent f'(x), [f]"(x)--the successive
differential coefficients of a limited function [f](x). As an example
of such formulae, consider the sine series (1); [f](x) is represented
by
_
2 n[pi]x /l n[pi]x
-- [Sigma] sin ------ | [f](x) sin ------ dx;
l l _/0 l
_
/l n[pi]x
on integration by parts we have | [f](x)sin ------ dx
_/0 l
_ _
l | n[pi]a |
= ---- | [f](+0) [+-] [f](l - 0) + [Sigma] cos ------ {[f]([alpha] + 0) - [f]([alpha] - 0)} |
n[pi] |_ l _|
_
l /l n[pi]x
+ ----- | [f]'(x) cos ------ dx
n[pi] _/0 l

where [alpha] represent the points where [f](x) is discontinuous.
Hence if f(x) is represented by the series [Sigma]a_n sin (n[pi]x/l),
and [f]'(x) by the series [Sigma]b_n cos (n[pi]x/l), we have the
relation
_ _
n[pi] 2 | n[pi][alpha] |
b_n = ----- [alpha]_n - -- | [f](+0) [+-] [f](l - 0) + [Sigma]cos ------------ {[f]([alpha] + 0) - [f](alpha - 0)} |
l l |_ l _|

hence only when the function is everywhere continuous, and [f](+0)
[f](l - 0) are both zero, is the series which represents [f]'(x)
obtained at once by differentiating that which represents [f](x). The
form of the coefficient [alpha]_n discloses the discontinuities of the
function and of its differential coefficients, for on continuing the
integration by parts we find
_ _
2 | n[pi][alpha] |
[alpha]_n = ----- | [f](+0) [+-] [f](l - 0) + [Sigma] cos ------------ {[f]([alpha] + 0) - [f]([alpha] - 0)} |
n[pi] |_ l _|
_ _
2l | n[pi][beta] |
+ --------- | [f]'(+0) [+-] [f]'(l - 0) + [Sigma] sin ----------- {[f]'([beta] + 0) - [f]'([beta] - 0)} | + &c.
n^2[pi]^2 |_ l _|

where [beta] are the points at which [f]'(x) is discontinuous.

HISTORY AND LITERATURE OF THE THEORY

The history of the theory of the representation of functions by series
of sines and cosines is of great interest in connexion with the
progressive development of the notion of an arbitrary function of a
real variable, and of the peculiarities which such a function may
possess; the modern views on the foundations of the infinitesimal
calculus have been to a very considerable extent formed in this
connexion (see FUNCTION). The representation of functions by these
series was first considered in the 18th century, in connexion with the
problem of a vibrating cord, and led to a controversy as to the
possibility of such expansions. In a memoir published in 1747
(_Memoirs of the Academy of Berlin_, vol. iii.) D'Alembert showed that
the ordinate y at any time t of a vibrating cord satisfies a
differential equation of the form [delta]^y/[delta]t^2 = a^2
[delta]^y/[delta]x^2, where x is measured along the undisturbed length
of the cord, and that with the ends of the cord of length l fixed, the
appropriate solution is y = [f](at + x) - [f](at - x), where [f] is a
function such that [f](x) = [f](x + 2l); in another memoir in the same
volume he seeks for functions which satisfy this condition. In the
year 1748 (_Berlin Memoirs_, vol. iv.) Euler, in discussing the
problem, gave [f](x) = [alpha] sin [pi]x/l + [beta] sin 2[pi]x/l + ...
as a particular solution, and maintained that every curve, whether
regular or irregular, must be representable in this form. This was
objected to by D'Alembert (1750) and also by Lagrange on the ground
that irregular curves are inadmissible. D. Bernoulli (_Berlin
Memoirs_, vol. ix., 1753) based a similar result to that of Euler on
physical intuition; his method was criticized by Euler (1753). The
question was then considered from a new point of view by Lagrange, in
a memoir on the nature and propagation of sound (_Miscellanea
Taurensia_, 1759; [_OE]uvres_, vol. i.), who, while criticizing
Euler's method, considers a finite number of vibrating particles, and
then makes the number of them infinite; he did not, however, quite
fully carry out the determination of the coefficients in Bernoulli's
Series. These mathematicians were hampered by the narrow conception of
a function, in which it is regarded as necessarily continuous; a
discontinuous function was considered only as a succession of several
different functions. Thus the possibility of the expansion of a broken
function was not generally admitted. The first cases in which rational
functions are expressed in sines and cosines were given by Euler
(_Subsidium calculi sinuum_, Novi Comm. Petrop., vol. v., 1754-1755),
who obtained the formulae

1/2 [phi] = sin [phi] - 1/2 sin 2[phi] + 1/3 sin 3[phi] ...

[pi]^2 [phi]^2
------ - ------- = cos [phi] - 1/4 cos 2[phi] + 1/9 cos 3[phi] ...
12 4

In a memoir presented to the Academy of St Petersburg in 1777, but not
published until 1798, Euler gave the method afterwards used by
Fourier, of determining the coefficients in the expansions; he
remarked that if [Phi] is expansible in the form
_ _
1 /[pi] 2 /[pi]
A + B cos[phi] + C cos 2[phi] + ..., then A = ---- | [Phi]d[phi], B = ---- | cos [phi]d[phi], &c.
[pi] _/ 0 [pi] _/ 0

The second period in the development of the theory commenced in 1807,
when Fourier communicated his first memoir on the Theory of Heat to
the French Academy. His exposition of the present theory is contained
in a memoir sent to the Academy in 1811, of which his great treatise
the _Theorie analytique de la chaleur_, published in 1822, is, in the
main, a reproduction. Fourier set himself to consider the
representation of a function given graphically, and was the first
fully to grasp the idea that a single function may consist of detached
portions given arbitrarily by a graph. He had an accurate conception
of the convergence of a series, and although he did not give a
formally complete proof that a function with discontinuities is
representable by the series, he indicated in particular cases the
method of procedure afterwards carried out by Dirichlet. As an
exposition of principles, Fourier's work is still worthy of careful
perusal by all students of the subject. Poisson's treatment of the
subject, which has been adopted in English works (see the _Journal de
l'ecole polytechnique_, vol. xi., 1820, and vol. xii., 1823, and also
his treatise, _Theorie de la chaleur_, 1835), depends upon the equality
_
/[pi] 1 - h^2
| [f]([alpha]) ------------------------------ d[alpha]
_/-[pi] 1 - 2h cos (x - [alpha]) + h^2

_ _
1 /[pi] 1 /[pi]
= ----- | [f]([alpha]) d[alpha] + ---- [Sigma]h^n | [f]([alpha]) cos n(x - [alpha]) d[alpha]
2[pi] _/-[pi] [pi] _/-[pi]

where 0 < h < 1; the limit of the integral on the left-hand side is
evaluated when h=1, and found to be 1/2 {[f](x + 0) + [f](x - 0)}, the
series on the right-hand side becoming Fourier's Series. The equality
of the two limits is then inferred. If the series is assumed to be
convergent when h = 1, by a theorem of Abel's its sum is continuous
with the sum for values of h less than unity, but a proof of the
convergency for h = 1 is requisite for the validity of Poisson's
proof; as Poisson gave no such proof of convergency, his proof of the
general theorem cannot be accepted. The deficiency cannot be removed
except by a process of the same nature as that afterwards applied by
Dirichlet. The definite integral has been carefully studied by Schwarz
(see two memoirs in his collected works on the integration of the
equation [delta]^2u/[delta]x^2 + [delta]^2u/[delta]y^2 = 0), who showed
that the limiting value of the integral depends upon the manner in
which the limit is approached. Investigations of Fourier's Series were
also given by Cauchy (see his "Memoire sur les developpements des
fonctions en series periodiques," _Mem. de l'Inst_., vol. vi., also
_Oeuvres completes_, vol. vii.); his method, which depends upon a use
of complex variables, was accepted, with some modification, as valid
by Riemann, but one at least of his proofs is no longer regarded as
satisfactory. The first completely satisfactory investigation is due
to Dirichlet; his first memoir appeared in _Crelle's Journal_ for
1829, and the second, which is a model of clearness, in Dove's
_Repertorium der Physik_. Dirichlet laid down certain definite
sufficient conditions in regard to the nature of a function which is
expansible, and found under these conditions the limiting value of the
sum of n terms of the series. Dirichlet's determination of the sum of
the series at a point of discontinuity has been criticized by Schlafli
(see _Crelle's Journal_, vol. lxxii.) and by Du Bois-Reymond (_Mathem.
Annalen_, vol. vii.), who maintained that the sum is really
indeterminate. Their objection appears, however, to rest upon a
misapprehension as to the meaning of the sum of the series; if x1 be
the point of discontinuity, it is possible to make x approach x1,
and n become indefinitely great, so that the sum of the series takes
any assigned value in a certain interval, whereas we ought to make x =
x1 first and afterwards n = [oo], and no other way of going to the
double limit is really admissible. Other papers by Dircksen (_Crelle_,
vol. iv.) and Bessel (_Astronomische Nachrichten_, vol. xvi.), on
similar lines to those by Dirichlet, are of inferior importance. Many
of the investigations subsequent to Dirichlet's have the object of
freeing a function from some of the restrictions which were imposed
upon it in Dirichlet's proof, but no complete set of necessary and
sufficient conditions as to the nature of the function has been
obtained. Lipschitz ("De explicatione per series trigonometricas,"
_Crelle's Journal_, vol. lxiii., 1864) showed that, under a certain
condition, a function which has an infinite number of maxima and
minima in the neighbourhood of a point is still expansible; his
condition is that at the point of discontinuity [beta], |[f]([beta] +
[delta]) -f([beta])| < B[delta]^[alpha] as [delta] converges to zero,
B being a constant, and a a positive exponent. A somewhat wider
condition is

{[f]([beta] + [delta]) - [f]([beta])} log [delta]) = 0,
[delta] = 0

for which Lipschitz's results would hold. This last condition is
adopted by Dini in his treatise (_Sopra la serie di Fourier_, &c.,
Pisa, 1880).

The modern period in the theory was inaugurated by the publication by
Riemann in 1867 of his very important memoir, written in 1854, _Uber
die Darstellbarkeit einer Function durch eine trigonometrische Reihe_.
The first part of his memoir contains a historical account of the work
of previous investigators; in the second part there is a discussion of
the foundations of the Integral Calculus, and the third part is mainly
devoted to a discussion of what can be inferred as to the nature of a
function respecting the changes in its value for a continuous change
in the variable, if the function is capable of representation by a
trigonometrical series. Dirichlet and probably Riemann thought that
all continuous functions were everywhere representable by the series;
this view was refuted by Du Bois-Reymond (_Abh. der Bayer. Akad._ vol.
xii. 2). It was shown by Riemann that the convergence or
non-convergence of the series at a particular point x depends only
upon the nature of the function in an arbitrarily small neighbourhood
of the point x. The first to call attention to the importance of the
theory of uniform convergence of series in connexion with Fourier's
Series was Stokes, in his memoir "On the Critical Values of the Sums
of Periodic Series" (_Camb. Phil. Trans._, 1847; _Collected Papers_,
vol. i.). As the method of determining the coefficients in a
trigonometrical series is invalid unless the series converges in
general uniformly, the question arose whether series with coefficients
other than those of Fourier exist which represent arbitrary functions.
Heine showed (_Crelle's Journal_, vol. lxxi., 1870, and in his
treatise _Kugelfunctionen_, vol. i.) that Fourier's Series is in
general uniformly convergent, and that if there is a uniformly
convergent series which represents a function, it is the only one of
the kind. G. Cantor then showed (_Crelle's Journal_, vols. lxxii.
lxxiii.) that even if uniform convergence be not demanded, there can
be but one convergent expansion for a function, and that it is that of
Fourier. In the _Math. Ann._ vol. v., Cantor extended his
investigation to functions having an infinite number of
discontinuities. Important contributions to the theory of the series
have been published by Du Bois-Reymond (_Abh. der Bayer. Akademie_,
vol. xii., 1875, two memoirs, also in Crelle's Journal, vols. lxxiv.
lxxvi. lxxix.), by Kronecker (_Berliner Berichte_, 1885), by O. Holder
(_Berliner Berichte_, 1885), by Jordan (_Comptes rendus_, 1881, vol.
xcii.), by Ascoli (_Math. Annal._, 1873, and _Annali di matematica_,
vol. vi.), and by Genocchi (_Atti della R. Acc. di Torino_, vol. x.,
1875). Hamilton's memoir on "Fluctuating Functions" (_Trans. R.I.A._,
vol. xix., 1842) may also be studied with profit in this connexion. A
memoir by Broden (_Math. Annalen_, vol. lii.) contains a good
investigation of some of the most recent results on the subject. The
scope of Fourier's Series has been extended by Lebesgue, who
introduced a conception of integration wider than that due to Riemann.
Lebesgue's work on Fourier's Series will be found in his treatise,
_Lecons sur les series trigonometriques_ (1906); also in a memoir,
"Sur les series trigonometriques," _Annales sc. de l'ecole normale
superieure_, series ii. vol. xx. (1903), and in a paper "Sur la
convergence des series de Fourier," _Math. Annalen_, vol. lxiv.
(1905).

AUTHORITIES.--The foregoing historical account has been mainly drawn
from A. Sachse's work, "Versuch einer Geschichte der Darstellung
willkurlicher Functionen einer Variabeln durch trigonometrische
Reihen," published in _Schlomilch's Zeitschrift fur Mathematik_,
Supp., vol. xxv. 1880, and from a paper by G.A. Gibson "On the History
of the Fourier Series" (_Proc. Ed. Math. Soc._ vol. xi.). Reiff's
_Geschichte der unendlichen Reihen_ may also be consulted, and also
the first part of Riemann's memoir referred to above. Besides Dini's
treatise already referred to, there is a lucid treatment of the
subject from an elementary point of view in C. Neumann's treatise,
_Uber die nach Kreis-, Kugel- und Cylinder-Functionen fortschreitenden
Entwickelungen_. Jordan's discussion of the subject in his _Cours
d'analyse_ is worthy of attention: an account of functions with
limited variation is given in vol. i.; see also a paper by Study in
the _Math. Annalen_, vol. xlvii. On the second mean-value theorem
papers by Bonnet (Brux. Memoires, vol. xxiii., 1849, _Lionville's
Journal_, vol. xiv., 1849), by Du Bois-Reymond (_Crelle's Journal_,
vol. lxxix., 1875), by Hankel (_Zeitschrift fur Math. und Physik_,
vol. xiv., 1869), by Meyer (_Math. Ann._, vol. vi., 1872) and by
Holder (_Gottinger Anzeigen_, 1894) may be consulted; the most general
form of the theorem has been given by Hobson (_Proc. London Math.
Soc._, Series II. vol. vii., 1909). On the theory of uniform
convergence of series, a memoir by W.F. Osgood (_Amer. Journal of
Math._ xix.) may be with advantage consulted. On the theory of series
in general, in relation to the functions which they can represent, a
memoir by Baire (_Annali di matematica_, Series III. vol. iii.) is of
great importance. Bromwich's _Theory of Infinite Series_ (1908)
contains much information on the general theory of series. Bocher's
"Introduction to the Theory of Fourier's Series," _Annals of Math._,
Series II. vol. vii., 1906, will be found useful. See also Carslaw's
_Introduction to the Theory of Fourier's Series and Integrals, and the
Mathematical Theory of the Conduction of Heat_ (1906). A full account
of the theory will be found in Hobson's treatise _On the Theory of
Functions of a Real Variable and on the Theory of Fourier's Series_
(1907). (E. W. H.)

FOURMIES, a town of northern France, in the department of Nord, on an affluent of the Sambre, 39 m. S.E. of Valenciennes by rail. Pop. (1906) 13,308. It is one of the chief centres in France for wool combing and spinning, and produces a great variety of cloths. The glass-works of Fourmies date from 1599, and were the first established in the north of France. Iron is worked in the vicinity, and there are important forges and foundries. Enamel-ware is also manufactured. In 1891 labour troubles brought about military intervention and consequent bloodshed. A board of trade arbitration and a school of commerce and industry are among the public institutions.

FOURMONT, ETIENNE (1683-1745), French orientalist, was born at Herbelai, near Saint Denis, on the 23rd of June 1683. He studied at the College Mazarin, Paris, and afterwards in the College Montaigu, where his attention was attracted to Oriental languages. Shortly after leaving the college he published a _Traduction du commentaire du Rabbin Abraham Aben Esra sur l'ecclesiast_e. In 1711 Louis XIV. appointed Fourmont to assist a young Chinese, Hoan-ji, in compiling a Chinese grammar. Hoan-ji died in 1716, and it was not until 1737 that Fourmont published _Meditationes Sinicae_ and in 1742 _Grammatica Sinica_. He also wrote _Reflexions critiques sur les histoires des anciens peuples_ (1735), and several dissertations printed in the _Memoires_ of the Academy of Inscriptions. He became professor of Arabic in the College de France in 1715. In 1713 he was elected a member of the Academy of Inscriptions, in 1738 a member of the Royal Society of London, and in 1742 a member of that of Berlin. He died at Paris on the 19th of December 1745.

His brother, Michel Fourmont (1690-1746), was also a member of the Academy of Inscriptions, and professor of the Syriac language in the Royal College, and was sent by the government to copy inscriptions in Greece.

An account of Etienne Fourmont's life and a catalogue of his works
will be found in the second edition (1747) of his _Reflexions
critiques_.

FOURNET, JOSEPH JEAN BAPTISTE XAVIER (1801-1869), French geologist and metallurgist, was born at Strassburg on the 15th of May 1801. He was educated at the Ecole des Mines at Paris, and after considerable experience as a mining engineer he was in 1834 appointed professor of geology at Lyons. He was a man of wide knowledge and extensive research, and wrote memoirs on chemical and mineralogical subjects, on eruptive rocks, on the structure of the Jura, the metamorphism of the Western Alps, on the formation of oolitic limestones, on kaolinization and on metalliferous veins. On metallurgical subjects also he was an acknowledged authority; and he published observations on the order of sulphurability of metals (_loi de Fournet_). He died at Lyons on the 8th of January 1869. His chief publications were: E_tudes sur les depots metalliferes_ (Paris, 1834); _Histoire de la dolomie_ (Lyons, 1847); _De l'extension des terrains houillers_ (1855); _Geologie lyonnaise_ (Lyons, 1861).

FOURNIER, PIERRE SIMON (1712-1768), French engraver and typefounder, was born at Paris on the 15th of September 1712. He was the son of a printer, and was brought up to his father's business. After studying drawing under the painter Colson, he practised for some time the art of wood-engraving, and ultimately turned his attention to the engraving and casting of types. He designed many new characters, and his foundry became celebrated not only in France, but in foreign countries. Not content with his practical achievements, he sought to stimulate public interest in his art by the production of various works on the subject. In 1737 he published his _Table des proportions qu'il faut observer entre les caracteres_, which was followed by several other technical treatises. In 1758 he assailed the title of Gutenberg to the honour awarded him as inventor of printing, claiming it for Schoffer, in his _Dissertation sur l'origine et les progres de l'art de graver en bois_. This gave rise to a controversy in which Schopflin and Baer were his opponents. Fournier's contributions to this debate were collected and reprinted under the title of _Traites historiques et critiques sur l'origine de l'imprimerie_. His principal work, however, was the _Manuel typographique_, which appeared in 2 vols. 8vo in 1764, the first volume treating of engraving and type-founding, the second of printing, with examples of different alphabets. It was the author's design to complete the work in four volumes, but he did not live to execute it. He died at Paris on the 8th of October 1768.

FOURNIER L'HERITIER, CLAUDE (1745-1825), French revolutionist, called "l'Americain," was born at Auzon (Haute-Loire) on the 21st of December 1745, the son of a poor weaver. He went to America to seek his fortune, and started at San Domingo an establishment for making _tafia_ (an inferior quality of rum), but lost his money in a fire. Returning to France he threw himself into the Revolution with enthusiasm, and specially distinguished himself by the active part he took in the organization of the popular armed force by means of which the most famous of the revolutionary _coups_ were effected. His influence was principally manifested in the insurrections of the 5th and 6th of October 1789, the 17th of July 1791, and the 20th of June and the 10th of August 1792. He was on bad terms with the majority of the politicians, and particularly with Marat, and spent a great part of his time in prison, all the governments regarding him as an agitator and accusing him of inciting to insurrection. Arrested for the first time for trying to force an entrance into the club of the Cordeliers, from which he had been expelled, he was released, but was in prison from the 12th of December 1793 to the 21st of September 1794, and again from the 9th of March 1795 to the 26th of October 1795. After the attempt on the First Consul in the rue Sainte-Nicaise he was deported to Guiana, but was allowed to return to France in 1809. In 1811, while under surveillance at Auxerre, he was accused of having provoked an _emeute_ against taxes known as the _droits reunis_ (afterwards called c_ontributions indirectes_), and was imprisoned in the Chateau d'If, where he remained till 1814. On the second restoration of the Bourbons Fournier was confined for about nine months in the prison of La Force. After 1816 he was left unmolested, turned royalist, and passed his last years in importuning the Restoration government for compensation for his lost property in San Domingo. He died in obscurity.

For further details see preface to F.A. Aulard's edition of Fournier's
_Memoires secrets_ (Paris, 1890), published by the Societe de
l'histoire de la Revolution.

FOURTOU, MARIE FRANCOIS OSCAR BARDY DE (1836-1897), French politician, was born at Riberac (Dordogne) on the 3rd of January 1836, and represented his native department in the National Assembly after the Franco-German War. There he proved a useful adherent to Thiers, who made him minister of public works in December 1872. He was minister of religion in the cabinet of May 18-24, 1873, being the only member of the Right included by Thiers in that short-lived ministry. As minister of education, religion and the fine arts in the reconstructed cabinet of the duc de Broglie he had used his administrative powers to further clerical ends, and as minister of the interior in Broglie's cabinet in 1877 he resumed the administrative methods of the Second Empire. With a well-known Bonapartist, Baron R.C.F. Reille, as his secretary, he replaced republican functionaries by Bonapartist partisans, reserving a few places for the Legitimists. In the general elections of that year he used the whole weight of officialdom to secure a majority for the Right, to support a clerical and reactionary programme. He accompanied Marshal MacMahon in his tour through southern France, and the presidential manifesto of September, stating that the president would rely solely on the Senate should the elections prove unfavourable, was generally attributed to Fourtou. In spite of these efforts the cabinet fell, and a commission was appointed to inquire into their unconstitutional abuse of power. Fourtou was unseated in consequence of the revelations made in the report of the commission. In the Chamber of Deputies Gambetta gave the lie direct to Fourtou's allegation that the republican party opposed every republican principle that was not antiquated. A duel was fought in consequence, but neither party was injured. He was re-elected to the chamber in 1879 and entered the Senate the next year. Failing to secure re-election to the Senate in 1885 he again entered the popular chamber as Legitimist candidate in 1889, but he took no further active part in politics. He died in Paris in 1897.

His works include _Histoire de Louis XVI_ (1840); _Histoire de Saint
Pie V_ (1845); _Mme Swetchine, sa vie et ses oeuvres_ (2 vols., 1859);
_La Question italienne_ (1860); _De la contre-revolution_ (1876); and
_Memoires d'un royaliste_ (2 vols., 1888).

FOUSSA, or FOSSA, the native name of _Cryptoprocta ferox_, a somewhat cat-like or civet-like mammal peculiar to Madagascar, where it is the largest carnivorous animal. It is about twice the size of a cat (5 ft. from nose to end of tail), with short close fur of nearly uniform pale brown. Little is known of its habits, except that it is nocturnal, frequently attacks and carries off goats, and especially kids, and shows great ferocity when wounded, on which account it is much dreaded by the natives. An example lived in the London zoological gardens for nearly fourteen years. See CARNIVORA.

FOWEY (usually pronounced _Foy_), a seaport and market-town in the Bodmin parliamentary division of Cornwall, England, on the Great Western railway, 25 m. by sea W. of Plymouth. Pop. (1901) 2258. It lies on the west shore of the picturesque estuary of the river Fowey, close to the water's edge, and sheltered by a screen of hills. Its church of St Nicholas is said to have been built in the 14th century, on the site of a still older edifice dedicated to St Finbar of Cork. It has a fine tower and late Norman doorway. Within are a priest's chamber over the porch, a handsome oak ceiling, a 15th-century pulpit, and some curious monuments and brasses. Place House, adjacent to the church, is a highly ornate Tudor building. A few ancient houses remain in the town. Deep-sea fishing is carried on; but the staple trade consists in the export of china clay and minerals, coal being imported. Fowey harbour, which is easy of access in clear weather, will admit large vessels at any state of the tide. St Catherine's Fort, dating from the days of Henry VIII. and now ruined, stands at the harbour's mouth, and once formed the main defence of the town. Opposite the town, and connected with it by Bodeneck Ferry, is the village of Polruan. Its main features are St Saviour's Chapel, with an ancient rood-stone, and the remains of Hall House, which was garrisoned during the civil wars of the 17th century.