can be formed in the same way as those of the first order. For
example, when there are two variables x, y, the first partial
derivatives ð[f]/ðx and ð[f]/ðy are functions of x and y, which we may
seek to differentiate partially with respect to x or y. The most
important theorem in relation to partial differential coefficients of
orders higher than the first is the theorem that the values of such
coefficients do not depend upon the order in which the
differentiations are performed. For example, we have the equation
ð /ð[f]\ ð /ð[f]\
-- ( ---- ) = -- ( ---- ) (i.)
ðx \ ðy / ðy \ ðx /
This theorem is not true without limitation. The conditions for its
validity have been investigated very completely by H. A. Schwarz (see
his _Ges. math. Abhandlungen_, Bd. 2, Berlin, 1890, p. 275). It is a
sufficient, though not a necessary, condition that all the
differential coefficients concerned should be continuous functions of
x, y. In consequence of the relation (i.) the differential
coefficients expressed in the two members of this relation are written
ð²f ð²f
---- or ----.
ðxðy ðyðx
The differential coefficient
ð^_n [f]
--------------,
ðx^p ðy^q ðz^r
in which p + g + r = n, is formed by differentiating p times with
respect to x, q times with respect to y, r times with respect to z,
the differentiations being performed in any order. Abbreviated
notations are sometimes used in such forms as
(p, q, r)
[f] or [f] .
x^p y^q z^r x, y, z
_Differentials_ of higher orders are introduced by the defining
equation
/ ð ð \ n
d^n [f] = ( dx -- + dy -- ) [f]
\ ðx ðy/
ð^n [f] ð^n [f]
= (dx)^n ------- + n(dx)^(n-1) dy ----------- + ...
ðx^n ðx^(n-1) ðy
in which the expression (dx·ð/ðx + dy·ð/ðy)^n is developed by the
binomial theorem in the same way as if dx·ð/ðx and dy·ð/ðy were
numbers, and (ð/ðx)^r·(ð/ðy)^(n-r) [f] is replaced by ð^n [f]/[ðx^r
ðy^(n-r)]. When there are more than two variables the multinomial
theorem must be used instead of the binomial theorem.
The problem of forming the second and higher differential coefficients
of _implicit functions_ can be solved at once by means of partial
differential coefficients, for example, if [f](x, y) = 0 is the
equation defining y as a function of x, we have
_ _
d²y /ð[f]\ -3 | /ð[f]\² ð²[f] ð[f] ð[f] ð²[f] /ð[f]\² ð²[f] |
--- = ( ---- ) | ( ---- ) ----- - 2 ---- · ---- · ----- + ( ---- ) ----- |.
dx² \ ðy / |_ \ ðy / ðx² ðx ðy ðxðy \ ðx / ðy² _|
The differential expression Xdx + Ydy, in which both X and Y are
functions of the two variables x and y, is a _total differential_ if
there exists a function [f] of x and y which is such that
ð[f]/ðx = X, ð[f]/ðy = Y.
When this is the case we have the relation
ðY/ðx = ðX/ðy. (ii.)
Conversely, when this equation is satisfied there exists a function
[f] which is such that
d[f] = Xdx + Ydy.
The expression Xdx + Ydy in which X and Y are connected by the
relation (ii.) is often described as a "perfect differential." The
theory of the perfect differential can be extended to functions of n
variables, and in this case there are ½n(n - 1) such relations as
(ii.).
In the case of a function of two variables x, y an abbreviated
notation is often adopted for differential coefficients. The function
being denoted by z, we write
ðz ðz ð²z ð²z ð²z
p, q, r, s, t for --, --, ---, ----, ---.
ðx ðy ðx² ðxðy ðy²
Partial differential coefficients of the second order are important in
geometry as expressing the curvature of surfaces. When a surface is
given by an equation of the form z = [f](x, y), the lines of curvature
are determined by the equation
{(l + q²)s - pqt} (dy)² + {(1 + q²)r - (1 + p²)t} dx dy
- {(1 + p²)s - pqr} (dx)² = 0,
and the principal radii of curvature are the values of R which satisfy
the equation
R²(rt - s²) - R{(1 + q²)r - 2pqs + (1 + p²)t} [root](1 + p² + q²)
+ (1 + p² + q²)² = 0.
Change of variables.
44. The problem of change of variables was first considered by Brook
Taylor in his _Methodus incrementorum_. In the case considered by
Taylor y is expressed as a function of z, and z as a function of x,
and it is desired to express the differential coefficients of y with
respect to x without eliminating z. The result can be obtained at once
by the rules for differentiating a product and a function of a
function. We have
dy dy dz
-- = -- · --,
dx dz dx
d²y dy d²z d²y /dz\²
--- = -- · --- + --- · ( -- ),
dx² dz dx² dz² \dx/
d³y dy d³z, d²y dz d²z, d³y /dz\³
--- = -- · --- + 3 --- · -- · --- + --- · ( -- ) ,
dx³ dz dx³ dz² dx dx² dz³ \dx/
. . . . . . .
The introduction of partial differential coefficients enables us to
deal with more general cases of change of variables than that
considered above. If u, v are new variables, and x, y are connected
with them by equations of the type
x = [f]1(u, v), y = [f]2(u, v), (i.)
while y is either an explicit or an implicit function of x, we have
the problem of expressing the differential coefficients of various
orders of y with respect to x in terms of the differential
coefficients of v with respect to u. We have
dy /ð[f]2 ð[f]2 dv \ / /ð[f]1 ð[f]1 dv \
-- = ( ----- + ----- -- ) / ( ----- + ----- -- )
dx \ ðu ðv du / / \ ðu ðv du /
by the rule of the total differential. In the same way, by means of
differentials of higher orders, we may express d²y/dx², and so on.
Equations such as (i.) may be interpreted as effecting a
_transformation_ by which a point (u, v) is made to correspond to a
point (x, y). The whole theory of transformations, and of functions,
or differential expressions, which remain invariant under groups of
transformations, has been studied exhaustively by Sophus Lie (see, in
particular, his _Theorie der Transformationsgruppen_, Leipzig,
1888-1893). (See also DIFFERENTIAL EQUATIONS and GROUPS).
A more general problem of change of variables is presented when it is
desired to express the partial differential coefficients of a function
V with respect to x, y, ... in terms of those with respect to u, v,
..., where u, v, ... are connected with x, y, ... by any functional
relations. When there are two variables x, y, and u, v are given
functions of x, y, we have
ðV ðV ðu ðV ðv
-- = -- -- + -- --,
ðx ðu ðx ðv ðx
ðV ðV ðu ðV ðv
-- = -- -- + -- --,
ðy ðu ðy ðv ðy
and the differential coefficients of higher orders are to be formed by
repeated applications of the rule for differentiating a product and
the rules of the type
ð ðu ð ðv ð
-- = -- -- + -- --.
ðx ðx ðu ðx ðx
When x, y are given functions of u, v, ... we have, instead of the
above, such equations as
ðV ðV ðx ðV ðy
-- = -- -- + -- --;
ðu ðx ðu ðy ðu
and ðV/ðx, ðV/ðy can be found by solving these equations, provided the
Jacobian ð(x, y) / ð(u, v) is not zero. The generalization of this
method for the case of more than two variables need not detain us.
In cases like that here considered it is sometimes more convenient not
to regard the equations connecting x, y with u, v as effecting a point
transformation, but to consider the loci u = const., v = const. as two
"families" of curves. Then in any region of the plane of (x, y) in
which the Jacobian ð(x, y) / d(u, v) does not vanish or become
infinite, any point (x, y) is uniquely determined by the values of u
and v which belong to the curves of the two families that pass through
the point. Such variables as u, v are then described as "curvilinear
coordinates" of the point. This method is applicable to any number of
variables. When the loci u = const., ... intersect each other at right
angles, the variables are "orthogonal" curvilinear coordinates.
Three-dimensional systems of such coordinates have important
applications in mathematical physics. Reference may be made to G.
Lamé, _Leçons sur les coordonnées curvilignes_ (Paris, 1859), and to
G. Darboux, _Leçons sur les coordonnées curvilignes et systèmes
orthogonaux_ (Paris, 1898).
When such a coordinate as u is connected with x and y by a functional
relation of the form [f](x, y, u) = 0 the curves u = const. are a
family of curves, and this family may be such that no two curves of
the family have a common point. When this is not the case the points
in which a curve [f](x, y, u) = 0 is intersected by a curve [f](x, y,
u + [Delta]u) = 0 tend to limiting positions as [Delta]u is diminished
indefinitely. The locus of these limiting positions is the "envelope"
of the family, and in general it touches all the curves of the family.
It is easy to see that, if u, v are the parameters of two families of
curves which have envelopes, the Jacobian ð(x, y) / ð(u, v) vanishes
at all points on these envelopes. It is easy to see also that at any
point where the reciprocal Jacobian ð(u, v) / ð(x, y) vanishes, a
curve of the family u touches a curve of the family v.
If three variables x, y, z are connected by a functional relation
[f](x, y, z) = 0, one of them, z say, may be regarded as an _implicit
function_ of the other two, and the partial differential coefficients
of z with respect to x and y can be formed by the rule of the total
differential. We have
ðz ð[f] / ð[f] ðz ð[f] / ð[f]
-- = - ---- / ----, -- = - ---- / ----;
ðx ðx / ðz ðy ðy / ðz
and there is no difficulty in proceeding to express the higher
differential coefficients. There arises the problem of expressing the
partial differential coefficients of x with respect to y and z in
terms of those of z with respect to x and y. The problem is known as
that of "changing the dependent variable." It is solved by applying
the rule of the total differential. Similar considerations are
applicable to all cases in which n variables are connected by fewer
than n equations.
Extension of Taylor's theorem.
45. Taylor's theorem can be extended to functions of several
variables. In the case of two variables the general formula, with a
remainder after n terms, can be written most simply in the form
[f](a + h, b + k) = [f](a, b) + d[f](a, b) + (1/2!) d²[f](a, b) + ...
1 1
+ -------- d^(n-1) [f](a, b) + -- d^n [f](a+[Theta]h, b + [theta]k),
(n - 1)! n!
in which
_ _
| / ð ð \r |
d^r [f](a, b) = | ( h -- + k -- ) [f](x, y) | ,
|_ \ ðx ðy / _| x=a, y=b
and
d^n [f](a + [Theta]h, b + [Theta]k) =
_ _
| / ð ð \n |
| ( h -- + k -- ) [f](x, y) |.
|_ \ ðx ðy/ _| x=a+[Theta]h, y=b+[Theta]k
The last expression is the remainder after n terms, and in it [Theta]
denotes some particular number between 0 and 1. The results for three
or more variables can be written in the same form. The extension of
Taylor's theorem was given by Lagrange (1797); the form written above
is due to Cauchy (1823). For the validity of the theorem in this form
it is necessary that all the differential coefficients up to the nth
should be continuous in a region bounded by x = a ± h, y = b ± k. When
all the differential coefficients, no matter how high the order, are
continuous in such a region, the theorem leads to an expansion of the
function in a multiple power series. Such expansions are just as
important in analysis, geometry and mechanics as expansions of
functions of one variable. Among the problems which are solved by
means of such expansions are the problem of maxima and minima for
functions of more than one variable (see MAXIMA and MINIMA).
Plane curves.
46. In treatises on the differential calculus much space is usually
devoted to the differential geometry of curves and surfaces. A few
remarks and results relating to the differential geometry of plane
curves are set down here.
(i.) If [psi] denotes the angle which the radius vector drawn from the
origin makes with the tangent to a curve at a point whose polar
coordinates are r, [Theta] and if p denotes the perpendicular from the
origin to the tangent, then
cos [psi] = dr/ds, sin [psi] = r d[Theta]/ds = p/r,
where ds denotes the element of arc. The curve may be determined by an
equation connecting p with r.
(ii.) The locus of the foot of the perpendicular let fall from the
origin upon the tangent to a curve at a point is called the _pedal_ of
the curve with respect to the origin. The angle [psi] for the pedal is
the same as the angle [psi] for the curve. Hence the (p, r) equation
of the pedal can be deduced. If the pedal is regarded as the primary
curve, the curve of which it is the pedal is the "negative pedal" of
the primary. We may have pedals of pedals and so on, also negative
pedals of negative pedals and so on. Negative pedals are usually
determined as envelopes.
(iii.) If [phi] denotes the angle which the tangent at any point makes
with a fixed line, we have
r² = p² + (dp/d[phi])².
(iv.) The "average curvature" of the arc [Delta]s of a curve between
two points is measured by the quotient
| [Delta][phi] |
| ------------ |
| [Delta]s |
where the upright lines denote, as usual, that the absolute value of
the included expression is to be taken, and [phi] is the angle which
the tangent makes with a fixed line, so that [Delta][phi] is the angle
between the tangents (or normals) at the points. As one of the points
moves up to coincidence with the other this average curvature tends to
a limit which is the "curvature" of the curve at the point. It is
denoted by
| d[phi] |
| ------ |
| ds |
Sometimes the upright lines are omitted and a rule of signs is
given:--Let the arc s of the curve be measured from some point along
the curve in a chosen sense, and let the normal be drawn towards that
side to which the curve is concave; if the normal is directed towards
the left of an observer looking along the tangent in the chosen sense
of description the curvature is reckoned positive, in the contrary
case negative. The differential d[phi] is often called the "angle of
contingence." In the 14th century the size of the angle between a
curve and its tangent seems to have been seriously debated, and the
name "angle of contingence" was then given to the supposed angle.
(v.) The curvature of a curve at a point is the same as that of a
certain circle which touches the curve at the point, and the "radius
of curvature" [rho] is the radius of this circle. We have 1/[rho] =
|d[phi]/ds|. The centre of the circle is called the "centre of
curvature"; it is the limiting position of the point of intersection
of the normal at the point and the normal at a neighbouring point,
when the second point moves up to coincidence with the first. If a
circle is described to intersect the curve at the point P and at two
other points, and one of these two points is moved up to coincidence
with P, the circle touches the curve at the point P and meets it in
another point; the centre of the circle is then on the normal. As the
third point now moves up to coincidence with P, the centre of the
circle moves to the centre of curvature. The circle is then said to
"osculate" the curve, or to have "contact of the second order" with it
at P.
(vi.) The following are formulae for the radius of curvature:--
1 | { /dy\² }-3/2 d²y |
----- = | { 1 + ( -- ) } --- |,
[rho] | { \dx/ } dx² |
| dr | | d²p |
[rho] = | r -- | = | p + ------- |.
| dp | | d[phi]² |
(vii.) The points at which the curvature vanishes are "points of
inflection." If P is a point of inflection and Q a neighbouring point,
then, as Q moves up to coincidence with P, the distance from P to the
point of intersection of the normals at P and Q becomes greater than
any distance that can be assigned. The equation which gives the
abscissae of the points in which a straight line meets the curve being
expressed in the form [f](x) = 0, the function [f](x) has a factor (x
- x0)³, where x0 is the abscissa of the point of inflection P, and the
line is the tangent at P. When the factor (x - x0) occurs (n + 1)
times in [f](x), the curve is said to have "contact of the nth order"
with the line. There is an obvious modification when the line is
parallel to the axis of y.
(viii.) The locus of the centres of curvature, or envelope of the
normals, of a curve is called the "evolute." A curve which has a given
curve as evolute is called an "involute" of the given curve. All the
involutes are "parallel" curves, that is to say, they are such that
one is derived from another by marking off a constant distance along
the normal. The involutes are "orthogonal trajectories" of the
tangents to the common evolute.
(ix.) The equation of an algebraic curve of the nth degree can be
expressed in the form u0 + u1 + u2 + ... + u_n = 0, where u0 is a
constant, and u_r is a homogeneous rational integral function of x, y
of the rth degree. When the origin is on the curve, u0 vanishes, and
u1 = 0 represents the tangent at the origin. If u1 also vanishes, the
origin is a double point and u2 = o represents the tangents at the
origin. If u2 has distinct factors, or is of the form a(y - p1x)(y -
p2x), the value of y on either branch of the curve can be expressed
(for points sufficiently near the origin) in a power series, which is
either
p1x + ½ q1x² + ..., or p2x + ½ q2X² + ...,
where q1, ... and q2, ... are determined without ambiguity. If p1 and
p2 are real the two branches have radii of curvature [rho]1, [rho]2
determined by the formulae
1 | | 1 | |
------ = |(1 + p1²)^{-3/2} q1 |, ------ = |(1 + p2²)^{-3/2} q2 |.
[rho]1 | | [rho]2 | |
When p1 and p2 are imaginary the origin is the real point of
intersection of two imaginary branches. In the real figure of the
curve it is an _isolated point_. If u2 is a square, a(y - px)², the
origin is a _cusp_, and in general there is not a series for y in
integral powers of x, which is valid in the neighbourhood of the
origin. The further investigation of cusps and multiple points belongs
rather to analytical geometry and the theory of algebraic functions
than to differential calculus.
(x.) When the equation of a curve is given in the form u0 + u1 + ... +
u_(n-1) + u_n = 0 where the notation is the same as that in (ix.), the
factors of u_n determine the directions of the _asymptotes_. If these
factors are all real and distinct, there is an asymptote corresponding
to each factor. If u_n = L1 L2 ... L_n, where L1, ... are linear in x,
y, we may resolve u_(n-1)/u_n into partial fractions according to the
formula
u_(n-1) A1 A2 A_n
------- = -- + -- + ... + ---,
u{n} L1 L2 L_n
and then L1 + A1 = 0, L2 + A2 = 0, ... are the equations of the
asymptotes. When a real factor of u_n is repeated we may have two
parallel asymptotes or we may have a "parabolic asymptote." Sometimes
the parallel asymptotes coincide, as in the curve x²(x² + y² - a²) =
a^4, where x = 0 is the only real asymptote. The whole theory of
asymptotes belongs properly to analytical geometry and the theory of
algebraic functions.
Integral calculus.
47. The formal definition of an integral, the theorem of the existence
of the integral for certain classes of functions, a list of classes of
"integrable" functions, extensions of the notion of integration to
functions which become infinite or indeterminate, and to cases in
which the limits of integration become infinite, the definitions of
multiple integrals, and the possibility of defining functions by means
of definite integrals--all these matters have been considered in
FUNCTION. The definition of integration has been explained in § 5
above, and the results of some of the simplest integrations have been
given in § 12. A few theorems relating to integrations have been noted
in §§ 34, 35, 36 above.
Methods of integration.
48. The chief methods for the evaluation of indefinite integrals are
the method of integration by parts, and the introduction of new
variables.
From the equation d(uv) = udv + vdu we deduce the equation
_ _
/ dv / du
| u -- dx = uv - | v -- dx,
_/ dx _/ dx
or, as it may be written
_ _ _ _
/ / / du / / \
| uw dx = u | w dx - | -- ( | w dx ) dx.
_/ _/ _/ dx \ _/ /
This is the rule of "integration by parts."
As an example we have
_ _
/ e^(ax) / e^(ax) / x 1 \
| xe^(ax) dx = x ------ - | ------ dx = ( --- - -- ) e^(ax).
_/ a _/ a \ a a² /
When we introduce a new variable z in place of x, by means of an
equation giving x in terms of z, we express [f](x) in terms of z. Let
[phi](z) denote the function of z into which [f](x) is transformed.
Then from the equation
dx
dx = -- dz
dz
we deduce the equation
_ _
/ / dx
| [f](x) dx = | [phi](z) -- dz.
_/ _/ dz
As an example, in the integral
_
/
| [root](1 - x²) dx
_/
put x = sin z; the integral becomes
_ _
/ /
| cos z · cos zdz = | ½(1 + cos 2z)dz = ½(z + ½ sin 2z) = ½(z + sin z cos z).
_/ _/
Integration in terms of elementary functions.
49. The indefinite integrals of certain classes of functions can be
expressed by means of a finite number of operations of addition or
multiplication in terms of the so-called "elementary" functions. The
elementary functions are rational algebraic functions, implicit
algebraic functions, exponentials and logarithms, trigonometrical and
inverse circular functions. The following are among the classes of
functions whose integrals involve the elementary functions only: (i.)
all rational functions; (ii.) all irrational functions of the form
[f](x, y), where [f] denotes a rational algebraic function of x and y,
and y is connected with x by an algebraic equation of the second
degree; (iii.) all rational functions of sin x and cos x; (iv.) all
rational functions of e^x; (v.) all rational integral functions of the
variables x, e^(ax), e^(bx), ... sin mx, cos mx, sin nx, cos nx, ...
in which a, b, ... and m, n, ... are any constants. The integration of
a rational function is generally effected by resolving the function
into partial fractions, the function being first expressed as the
quotient of two rational integral functions. Corresponding to any
simple root of the denominator there is a logarithmic term in the
integral. If any of the roots of the denominator are repeated there
are rational algebraic terms in the integral. The operation of
resolving a fraction into partial fractions requires a knowledge of
the roots of the denominator, but the algebraic part of the integral
can always be found without obtaining all the roots of the
denominator. Reference may be made to C. Hermite, _Cours d'analyse_,
Paris, 1873. The integration of other functions, which can be
integrated in terms of the elementary functions, can usually be
effected by transforming the functions into rational functions,
possibly after preliminary integrations by parts. In the case of
rational functions of x and a radical of the form [root](ax² + bx + c)
the radical can be reduced by a linear substitution to one of the
forms [root](a² - x²), [root](x² - a²), [root](x² + a²). The
substitutions x = a sin [theta], x = a sec [theta], x = a tan [theta]
are then effective in the three cases. By these substitutions the
subject of integration becomes a rational function of sin [theta] and
cos [theta], and it can be reduced to a rational function of t by the
substitution tan ½[theta] = t. There are many other substitutions by
which such integrals can be determined. Sometimes we may have
information as to the functional character of the integral without
being able to determine it. For example, when the subject of
integration is of the form (ax^4 + bx³ + cx² + dx + e)^-½ the integral
cannot be expressed explicitly in terms of elementary functions. Such
integrals lead to new functions (see FUNCTION).
Methods of reduction and substitution for the evaluation of indefinite
integrals occupy a considerable space in text-books of the integral
calculus. In regard to the functional character of the integral
reference may be made to G. H. Hardy's tract, _The Integration of
Functions of a Single Variable_ (Cambridge, 1905), and to the memoirs
there quoted. A few results are added here
_
/
(i.) | (x² + a) - ½ dx = log {x + (x² + a)^½ }.
_/
_
/ dx
(ii.) | -----------------------------
_/ (x - p) [root](ax² + 2bx + c)
can be evaluated by the substitution x - p = 1/z, and
_
/ dx
| ---------------------------------
_/ (x - p)^{n} [root](ax² + 2bx + c)
can be deduced by differentiating (n - 1) times with respect to p.
_
/ (Hx + K)dx
(iii.) | ------------------------------------------------------
_/ ([alpha]x² + 2[beta]x + [gamma]) [root](ax² + 2bx + c)
can be reduced by the substitution y² = (ax² + 2bx + c)/([alpha]x² +
2[beta]x + [gamma]) to the form
_ _
/ dy / dy
A | ---------------------- + B | ----------------------
_/ [root]([lambda]1 - y²) _/ [root](y² - [lambda]2)
where A and B are constants, and [lambda]1 and [lambda]2 are the two
values of [lambda] for which (a - [lambda][alpha])x² + 2(b -
[lambda][beta])x + c - [lambda][gamma] is a perfect square (see A. G.
Greenhill, _A Chapter in the Integral Calculus_, London, 1888).
(iv.) [f]x^m (ax^n + b)^p dx, in which m, n, p are rational, can be
reduced, by putting ax^n = bt, to depend upon [f]t^q (1 + t)^p dt. If
p is an integer and q a fraction r/s, we put t = u^s. If q is an
integer and p = r/s we put 1 + t = u^s. If p + q is an integer and p =
r/s we put 1 + t = tu^s. These integrals, called "binomial integrals,"
were investigated by Newton (_De quadratura curvarum_).
_ _
/ dx x / dx
(v.) | ----- = log tan ---, (vi.) | ----- = log (tan x + sec x).
_/ sin x 2 _/ cos x
(vii.) [f] e^(ax) sin (bx + [alpha]) dx = (a² + b²)^-1 e^(ax){a sin
(bx + [alpha]) - b cos (bx + [alpha])}.
(viii.) [f] sin^m x cos^n x dx can be reduced by differentiating a
function of the form sin^p x cos^q x;
d sin x 1 q sin² x 1 - q q
e.g. -- ------- = ----------- + ----------- = ----------- + -----------.
dx cos^q x cos^(q-1) x cos^(q+1) x cos^(q-1) x cos^(q+1) x
Hence
_ _
/ dx sin x n - 2 / dx
| ------- = ------------------- + ----- | -----------.
_/ cos^n x (n - 1) cos^(n-1) x n - 1 _/ cos^(n-2) x
_ _
/ ½[pi] / ½[pi]
(ix.) | sin^(2n) x dx = | cos^(2n) x dx =
_/ 0 _/ 0
1·3 ... (2n - 1) [pi]
---------------- · ----, (n an integer).
2·4 ... 2n 2
_ _
/ ½[pi] / ½[pi]
(x.) | sin^(2n+1) x dx = | cos^(2n+1) x dx =
_/ 0 _/ 0
2·4 ... (2n)
--------------, (n an integer).
3·5 ... (2n+1)
_
/ dx
(xi.) | --------------- can be reduced by one of the substitutions
_/ (1 + e cos x)^n
e + cos x e + cos x
cos [phi] = -----------, cosh u = -----------,
1 + e cos x 1 + e cos x
of which the first or the second is to be employed according as e < or > 1.
New transcendents.
50. Among the integrals of transcendental functions which lead to new
transcendental functions we may notice
_ _
/ x dx / log x e^z
| ----- or | --- dz,
_/ 0 log x´ _/ -x z
called the "logarithmic integral," and denoted by "Li x," also the
integrals
_ _
/ x sin x / x cos x
| ----- dx and | ----- dx,
_/ 0 x _/ [oo] x
called the "sine integral" and the "cosine integral," and denoted by
"Si x" and "Ci x," also the integral
_
/ x
| e^-x² dx
_/ 0
called the "error-function integral," and denoted by "Erf x." All
these functions have been tabulated (see TABLES, MATHEMATICAL).
Eulerian integrals.
51. New functions can be introduced also by means of the definite
integrals of functions of two or more variables with respect to one of
the variables, the limits of integration being fixed. Prominent among
such functions are the Beta and Gamma functions expressed by the
equations
_
/ 1
B(l, m) = | x^(l-1) (1 - x)^(m-1) dx,
_/ 0
_
/ [oo]
[Gamma](n) = | e^-t t^(n-1) dt.
_/ 0
When n is a positive integer [Gamma](n + 1) = n!. The Beta function
(or "Eulerian integral of the first kind") is expressible in terms of
Gamma functions (or "Eulerian integrals of the second kind") by the
formula
B(l, m)·[Gamma](l+m) = [Gamma](l)·[Gamma](m).
The Gamma function satisfies the difference equation
[Gamma](x + 1) = x [Gamma](x),
and also the equation
[Gamma](x)·[Gamma](1-x) = [pi]/sin (x[pi]),
with the particular result
[Gamma](½)= [root][pi].
The number
_ _
| d |
- | -- {log [Gamma](1 + x)} | , or -[Gamma]´(1),
|_ dx _|x=0
is called "Euler's constant," and is equal to the limit
_ _
| / \ |
lim. | ( 1 + ½ + 1/3 + ... + 1/n ) - log n |;
n=[oo] |_ \ / _|
its value to 15 decimal places is 0.577 215 664 901 532.
The function log [Gamma](1 + x) can be expanded in the series
/ x[pi] \
log [Gamma](1 + x) = ½ log ( --------- )
\ sin x[pi] /
1 + x
- ½ log ----- + {1 + [Gamma]´(1)} x
1 - x
- 1/3 (S3 - 1)x³ - 1/5 (S5 - 1)x^5 - ...,
where
1 1
S_(2r+1) = 1 + -------- + -------- + ...,
2^(2r+1) 3^(2r+1)
and the series for log [Gamma](1 + x) converges when x lies between -
1 and 1.
Definite integrals.
52. Definite integrals can sometimes be evaluated when the limits of
integration are some particular numbers, although the corresponding
indefinite integrals cannot be found. For example, we have the result
_
/ 1
| (1 - x²)^-½ log x dx = -½ [pi] log 2,
_/ 0
although the indefinite integral of (1 - x²)^-½ log x cannot be found.
Numbers of definite integrals are expressible in terms of the
transcendental functions mentioned in § 50 or in terms of Gamma
functions. For the calculation of definite integrals we have the
following methods:--
(i.) Differentiation with respect to a parameter.
(ii.) Integration with respect to a parameter.
(iii.) Expansion in infinite series and integration term by term.
(iv.) Contour integration.
The first three methods involve an interchange of the order of two
limiting operations, and they are valid only when the functions
satisfy certain conditions of continuity, or, in case the limits of
integration are infinite, when the functions tend to zero at infinite
distances in a sufficiently high order (see FUNCTION). The method of
contour integration involves the introduction of complex variables
(see FUNCTION: § _Complex Variables_).
A few results are added
_
/ [oo] x^(a-1) [pi]
(i.) | ------- dx = ---------, (1 > a > 0),
_/ 0 1 + x sin a[pi]
_
/ [oo] x^(a-1) - x^(b-1)
(ii.) | ----------------- dx = [pi](cot a[pi] - cot b[pi]), (0 < a or b < 1),
_/ 0 1 - x
_
/ [oo] x^(a-1) log x [pi]²
(iii.) | ------------ dx = ----------, (a > 1),
_/ 0 x - 1 sin² a[pi]
_
/ [oo]
(iv.) | x²·cos 2x·e^-x² dx = -¼ e^-1 [root][pi],
_/ 0
_
/ 1 1 - x² dx [pi]
(v.) | ------- ----- = log tan ----,
_/ 0 1 + x^4 log x 8
_
/ [oo] sin mx / 1 1 1 \
(vi.) | -------------- dx = ½ ( ------- - --- + --- ),
_/ 0 e^(2[pi]x) - 1 \ e^m - 1 m 2 /
_
/ [pi]
(vii.) | log(1 - 2[alpha] cos x + [alpha]²) dx = 0
_/ 0
or 2[pi]log [alpha] according as [alpha] < or > 1,
_
/ [oo] sin x
(viii.) | ----- dx = ½[pi],
_/ 0 x
_
/ [oo] cos ax
(ix.) | ------- dx = ½[pi]b^-1 e^(-ab),
_/ 0 x² + b²
_
/ [oo] cos ax - cos bx
(x.) | --------------- dx = ½[pi](b - a),
_/ 0 x²
_
/ [oo] cos ax - cos bx b
(xi.) | --------------- dx = log ---,
_/ 0 x a
_
/ [oo] cos x - e ^(-mx)
(xii.) | ---------------- dx = log m,
_/ 0 x
_
/ [oo]
(xiii.) | e^(-x²+2ax) dx = [root][pi].e^(a2),
_/ -[oo]
_ _
/ [oo] / [oo]
(xiv.) | x^-½ sin x dx = | x^-½ cos x dx = [root](½[pi]),
_/ 0 _/ 0
Multiple Integrals.
53. The meaning of integration of a function of n variables through a
domain of the same number of dimensions is explained in the article
FUNCTION. In the case of two variables x, y we integrate a function
[f](x, y) over an area; in the case of three variables x, y, z we
integrate a function [f](x, y, z) through a volume. The integral of a
function [f](x, y) over an area in the plane of (x, y) is denoted by
_ _
/ /
| | [f](x, y) dx dy.
_/_/
The notation refers to a method of evaluating the integral. We may
suppose the area divided into a very large number of very small
rectangles by lines parallel to the axes. Then we multiply the value
of [f] at any point within a rectangle by the measure of the area of
the rectangle, sum for all the rectangles, and pass to a limit by
increasing the number of rectangles indefinitely and diminishing all
their sides indefinitely. The process is usually effected by summing
first for all the rectangles which lie in a strip between two lines
parallel to one axis, say the axis of y, and afterwards for all the
strips. This process is equivalent to integrating [f](x, y) with
respect to y, keeping x constant, and taking certain functions of x as
the limits of integration for y, and then integrating the result with
respect to x between constant limits. The integral obtained in this
way may be written in such a form as
_ _
/ b { / [f]2(x) }
| dx { | [f](x, y) dy },
_/ a { _/ [f]1(x) }
and is called a "repeated integral." The identification of a surface
integral, such as [int][int][f](x, y)dxdy, with a repeated integral
cannot always be made, but implies that the function satisfies certain
conditions of continuity. In the same way volume integrals are usually
evaluated by regarding them as repeated integrals, and a volume
integral is written in the form
_ _ _
/ / /
| | | [f](x, y, z) dx dy dz.
_/_/_/
Integrals such as surface and volume integrals are usually called
"multiple integrals." Thus we have "double" integrals, "triple"
integrals, and so on. In contradistinction to multiple integrals the
ordinary integral of a function of one variable with respect to that
variable is called a "simple integral."
Surface Integrals.
A more general type of surface integral may be defined by taking an
arbitrary surface, with or without an edge. We suppose in the first
place that the surface is closed, or has no edge. We may mark a large
number of points on the surface, and draw the tangent planes at all
these points. These tangent planes form a polyhedron having a large
number of faces, one to each marked point; and we may choose the
marked points so that all the linear dimensions of any face are less
than some arbitrarily chosen length. We may devise a rule for
increasing the number of marked points indefinitely and decreasing the
lengths of all the edges of the polyhedra indefinitely. If the sum of
the areas of the faces tends to a limit, this limit is the area of the
surface. If we multiply the value of a function [f] at a point of the
surface by the measure of the area of the corresponding face of the
polyhedron, sum for all the faces, and pass to a limit as before, the
result is a surface integral, and is written
_ _
/ /
| | [f] dS.
_/_/
Line Integrals.
The extension to the case of an open surface bounded by an edge
presents no difficulty. A line integral taken along a curve is defined
in a similar way, and is written
_
/
| [f] ds
_/
where ds is the element of arc of the curve (§ 33). The direction
cosines of the tangent of a curve are dx/ds, dy/ds, dz/ds, and line
integrals usually present themselves in the form
_ _
/ / dx dy dz \ /
| ( u -- + v -- + w -- ) ds or | (u dx + v dy + w dz).
_/ \ ds ds ds / _/ s
In like manner surface integrals usually present themselves in the
form
_ _
/ /
| | (l[xi] + m[eta] + n[zeta]) dS
_/_/
where l, m, n are the direction cosines of the normal to the surface
drawn in a specified sense.
The area of a bounded portion of the plane of (x, y) may be expressed
either as
_
/
½ | (x dy - y dx),
_/
or as
_ _
/ /
| | dx dy,
_/_/
the former integral being a line integral taken round the boundary of
the portion, and the latter a surface integral taken over the area
within this boundary. In forming the line integral the boundary is
supposed to be described in the positive sense, so that the included
area is on the left hand.
Theorems of Green and Stokes.
53_a_. We have two theorems of transformation connecting volume
integrals with surface integrals and surface integrals with line
integrals. The first theorem, called "Green's theorem," is expressed
by the equation
_ _ _ _ _
/ / / / ð[xi] ð[eta] ð[zeta]\ / /
| | | ( ----- + ------ + ------- )dx dy dz = | | (l[xi] + m[eta] + n[zeta]) dS,
_/_/_/ \ ðx ðy ðz / _/_/
where the volume integral on the left is taken through the volume
within a closed surface S, and the surface integral on the right is
taken over S, and l, m, n denote the direction cosines of the normal
to S drawn outwards. There is a corresponding theorem for a closed
curve in two dimensions, viz.,
_ _ _
/ / / ð[xi] ð[eta]\ / / dy dx \
| | ( ----- + ------ ) dx dy = | ( [xi] -- - [eta] -- ) ds,
_/_/ \ ðx ðy / _/ \ ds ds /
the sense of description of s being the positive sense. This theorem
is a particular case of a more general theorem called "Stokes's
theorem." Let s denote the edge of an open surface S, and let S be
covered with a network of curves so that the meshes of the network are
nearly plane, then we can choose a sense of description of the edge of
any mesh, and a corresponding sense for the normal to S at any point
within the mesh, so that these senses are related like the directions
of rotation and translation in a right-handed screw. This convention
fixes the sense of the normal (l, m, n) at any point on S when the
sense of description of s is chosen. If the axes of x, y, z are a
right-handed system, we have Stokes's theorem in the form
_ _ _
/ / / { /ðw ðv\ /ðu ðw\ /ðv ðu\ }
| (u dx + v dy + w dz) = | | { l( -- - -- ) + m( -- - -- ) + n( -- - -- ) }dS,
_/ s _/_/ { \ðy ðz/ \ðz ðx/ \ðx ðy/ }
where the integral on the left is taken round the curve s in the
chosen sense. When the axes are left-handed, we may either reverse the
sense of l, m, n and maintain the formula, or retain the sense of l,
m, n and change the sign of the right-hand member of the equation. For
the validity of the theorems of Green and Stokes it is in general
necessary that the functions involved should satisfy certain
conditions of continuity. For example, in Green's theorem the
differential coefficients ð[xi]/ðx, ð[eta]/ðy, ð[zeta]/ðz must be
continuous within S. Further, there are restrictions upon the nature
of the curves or surfaces involved. For example, Green's theorem, as
here stated, applies only to simply-connected regions of space. The
correction for multiply-connected regions is important in several
physical theories.
Change of Variables in a Multiple Integral.
54. The process of changing the variables in a multiple integral, such
as a surface or volume integral, is divisible into two stages. It is
necessary in the first place to determine the differential element
expressed by the product of the differentials of the first set of
variables in terms of the differentials of the second set of
variables. It is necessary in the second place to determine the limits
of integration which must be employed when the integral in terms of
the new variables is evaluated as a repeated integral. The first part
of the problem is solved at once by the introduction of the Jacobian.
If the variables of one set are denoted by x1, x2, ..., x_n, and those
of the other set by u1, u2, ..., u_n, we have the relation
ð(x1, x2, ..., x_n)
dx1 dx2 ...dx_n = ------------------- du1 du2 ... du_n.
ð(u1, u2, ..., u_n)
In regard to the second stage of the process the limits of integration
must be determined by the rule that the integration with respect to
the second set of variables is to be taken through the same domain as
the integration with respect to the first set.
For example, when we have to integrate a function [f](x, y) over the
area within a circle given by x² + y² = a², and we introduce polar
coordinates so that x = r cos [theta], y = r sin [theta], we find that
r is the value of the Jacobian, and that all points within or on the
circle are given by a [>=] r [>=] o, 2[pi][>=][theta][>=]o, and we have
_ _ _ _
/ a / [root](a²-x²) / a /2[pi]
| dx | [f](x, y) dy = | dr | f(r cos [theta], r sin [theta]) r d[theta].
_/-a _/-[root](a²-x²) _/ 0 _/ 0
If we have to integrate over the area of a rectangle a [>=] x [>=] 0,
b [>=] y [>=] 0, and we transform to polar coordinates, the integral
becomes the sum of two integrals, as follows:--
_ _ _ _
/a / b /tan^-1 b/a /a sec [theta]
| dx | [f](x, y) dy = | d[theta] | [f](r cos [theta], r sin [theta]) r dr
_/0 _/ 0 _/ 0 _/0
_ _
/ ½[pi] /b cosec [theta]
+ | d[theta] | [f](r cos [theta], r sin [theta]) r dr.
_/tan^-1 b/a _/ 0
55. A few additional results in relation to line integrals and
multiple integrals are set down here.
Line Integrals and Multiple Integrals.
(i.) Any simple integral can be regarded as a line-integral taken
along a portion of the axis of x. When a change of variables is made,
the limits of integration with respect to the new variable must be
such that the domain of integration is the same as before. This
condition may require the replacing of the original integral by the
sum of two or more simple integrals.
(ii.) The line integral of a perfect differential of a one-valued
function, taken along any closed curve, is zero.
(iii.) The area within any plane closed curve can be expressed by
either of the formulae
_ _
/ /
| ½ r² d[theta] or | ½ p ds,
_/ _/
where r, [theta] are polar coordinates, and p is the perpendicular
drawn from a fixed point to the tangent. The integrals are to be
understood as line integrals taken along the curve. When the same
integrals are taken between limits which correspond to two points of
the curve, in the sense of line integrals along the arc between the
points, they represent the area bounded by the arc and the terminal
radii vectores.
(iv.) The volume enclosed by a surface which is generated by the
revolution of a curve about the axis of x is expressed by the formula
_
/
[pi] | y² dx,
_/
and the area of the surface is expressed by the formula
_
/
2[pi] | y ds,
_/
where ds is the differential element of arc of the curve. When the
former integral is taken between assigned limits it represents the
volume contained between the surface and two planes which cut the axis
of x at right angles. The latter integral is to be understood as a
line integral taken along the curve, and it represents the area of the
portion of the curved surface which is contained between two planes at
right angles to the axis of x.
(v.) When we use curvilinear coordinates [xi], [eta] which are
conjugate functions of x, y, that is to say are such that
ð[xi]/ðx = ð[eta]/ðy and ð[xi]/ðy = -ð[eta]/ðx,
the Jacobian ð([xi], [eta])/ð(x, v) can be expressed in the form
/ð[xi]\² /ð[eta]\²
( ----- ) + ( ------ ),
\ ðx / \ ðx /
and in a number of equivalent forms. The area of any portion of the
plane is represented by the double integral
_ _
/ /
| | J^-1 d[xi] d[eta],
_/_/
where J denotes the above Jacobian, and the integration is taken
through a suitable domain. When the boundary consists of portions of
curves for which [xi] = const., or [eta] = const., the above is
generally the simplest way of evaluating it.
(vi.) The problem of "rectifying" a plane curve, or finding its
length, is solved by evaluating the integral
_
/ { /dy\² }½
| { 1 + ( -- ) } dx,
_/ { \dx/ }
or, in polar coordinates, by evaluating the integral
_
/ { / dr \² }½
| { r² + ( -------- ) } d[theta].
_/ { \d[theta]/ }
In both cases the integrals are line integrals taken along the curve.
(vii.) When we use curvilinear coordinates [xi], [eta] as in (v.)
above, the length of any portion of a curve [xi] = const. is given by
the integral
_
/
| J^-½ d[eta]
_/
taken between appropriate limits for [eta]. There is a similar formula
for the arc of a curve [eta] = const.
(viii.) The area of a surface z = [f](x, y) can be expressed by the
formula
_ _
/ / { /ðz\² /ðz\² }½
| | { 1 + ( -- ) + ( -- ) } dx dy.
_/_/ { \ðx/ \ðy/ }
When the coordinates of the points of a surface are expressed as
functions of two parameters u, v, the area is expressed by the
formula
_ _ _ _
/ / | { ð(y, z) }² { ð(z, x) }² { ð(x, y) }² |½
| | | { ------- } + { ------- } + { ------- } | du dv.
_/_/ |_ { ð(u, v) } { ð(u, v) } { ð(u, v) } _|
When the surface is referred to three-dimensional polar coordinates r,
[theta], [phi] given by the equations
x = r sin [theta] cos [phi], y = r sin [theta] sin [phi],
z = r cos [theta],
and the equation of the surface is of the form r = [f]([theta],
[phi]), the area is expressed by the formula
_ _ _ _
/ / | { / ðr \² } / ðr \² |½
| | r | { r² + ( -------- ) } sin² [theta] + ( ------ ) | d[theta] d[phi].
_/_/ |_ { \ð[theta]/ } \ð[phi]/ _|
The surface integral of a function of ([theta], [phi]) over the
surface of a sphere r = const. can be expressed in the form
_ _
/2[pi] /[pi]
| d[phi] | F([theta], [phi]) r² sin [theta] d[theta].
_/ 0 _/ 0
In every case the domain of integration must be chosen so as to
include the whole surface.
(ix.) In three-dimensional polar coordinates the Jacobian
ð(x, y, z)
-------------------- = r² sin [theta]
ð(r, [theta], [phi])
The volume integral of a function F (r, [theta], [phi]) through the
volume of a sphere r = a is
_ _ _
/ a /2[pi] /[pi]
| dr | d[phi] | F(r, [theta], [phi]) r² sin [theta] d[theta].
_/ 0 _/ 0 _/ 0
(x.) Integrations of rational functions through the volume of an
ellipsoid x²/a² + y²/b² + z²/c² = 1 are often effected by means of a
general theorem due to Lejeune Dirichlet (1839), which is as follows:
when the domain of integration is that given by the inequality
/x1\[alpha]1 /x2\^[alpha]2 /x_n\[alpha]_n
( -- ) + ( -- ) + ... + ( --- ) [<=] 1
\a1/ \a2/ \a_n/
where the a's and [alpha]'s are positive, the value of the integral
_ _
/ /
| | ... x1^(n1-1)·x2^(n2-1) ... dx1 dx2 ...
_/_/
a1^(n1) a2^(n2) ... [Gamma] (n1/[alpha]1) [Gamma] (n2/[alpha]2)
is --------------------- ---------------------------------------------.
[alpha]1 [alpha]2 ... [Gamma](1 + n1/[alpha]1 + n2/[alpha]2 + ... )
If, however, the object aimed at is an integration through the volume
of an ellipsoid it is simpler to reduce the domain of integration to
that within a sphere of radius unity by the transformation x = a[xi],
y = b[eta], z = c[zeta], and then to perform the integration through
the sphere by transforming to polar coordinates as in (ix).
Approximate and Mechanical Integration.
56. Methods of approximate integration began to be devised very early.
Kepler's practical measurement of the focal sectors of ellipses (1609)
was an approximate integration, as also was the method for the
quadrature of the hyperbola given by James Gregory in the appendix to
his _Exercitationes geometricae_ (1668). In Newton's _Methodus
differentialis_ (1711) the subject was taken up systematically.
Newton's object was to effect the approximate quadrature of a given
curve by making a curve of the type
y = a0 + a1x + a2x² + ... + a_n x^n
pass through the vertices of (n + 1) equidistant ordinates of the
given curve, and by taking the area of the new curve so determined as
an approximation to the area of the given curve. In 1743 Thomas
Simpson in his _Mathematical Dissertations_ published a very
convenient rule, obtained by taking the vertices of three consecutive
equidistant ordinates to be points on the same parabola. The distance
between the extreme ordinates corresponding to the abscissae x = a and
x = b is divided into 2n equal segments by ordinates y1, y2, ...
y(2n-1), and the extreme ordinates are denoted by y0, y(2n). The
vertices of the ordinates y0, y1, y2 lie on a parabola with its axis
parallel to the axis of y, so do the vertices of the ordinates y2, y3,
y4, and so on. The area is expressed approximately by the formula
{(b - a)/6n} [y0 + y_(2n) + 2 (y2 + y4 + ... + y_(2n-2))
+ 4(y1 + y3 + ... + y_(2n-1)],
which is known as Simpson's rule. Since all simple integrals can be
represented as areas such rules are applicable to approximate
integration in general. For the recent developments reference may be
made to the article by A. Voss in _Ency. d. Math. Wiss._, Bd. II., A.
2 (1899), and to a monograph by B. P. Moors, _Valeur approximative
d'une intégrale définie_ (Paris, 1905).
Many instruments have been devised for registering mechanically the
areas of closed curves and the values of integrals. The best known are
perhaps the "planimeter" of J. Amsler (1854) and the "integraph" of
Abdank-Abakanowicz (1882).
BIBLIOGRAPHY.--For historical questions relating to the subject the
chief authority is M. Cantor, _Geschichte d. Mathematik_ (3 Bde.,
Leipzig, 1894-1901). For particular matters, or special periods, the
following may be mentioned: H. G. Zeuthen, _Geschichte d. Math. im
Altertum u. Mittelalter_ (Copenhagen, 1896) and _Gesch. d. Math. im
XVI. u. XVII. Jahrhundert_ (Leipzig, 1903); S. Horsley, _Isaaci
Newtoni opera quae exstant omnia_ (5 vols., London, 1779-1785); C. I.
Gerhardt, _Leibnizens math. Schriften_ (7 Bde., Leipzig, 1849-1863);
Joh. Bernoulli, _Opera omnia_ (4 Bde., Lausanne and Geneva, 1742).
Other writings of importance in the history of the subject are cited
in the course of the article. A list of some of the more important
treatises on the differential and integral calculus is appended. The
list has no pretensions to completeness; in particular, most of the
recent books in which the subject is presented in an elementary way
for beginners or engineers are omitted.--L. Euler, _Institutiones
calculi differentialis_ (Petrop., 1755) and _Institutiones calculi
integralis_ (3 Bde., Petrop., 1768-1770); J. L. Lagrange, _Leçons sur
le calcul des fonctions_ (Paris, 1806, _Oeuvres_, t. x.), and _Théorie
des fonctions analytiques_ (Paris, 1797, 2nd ed., 1813, _Oeuvres_, t.
ix.); S. F. Lacroix, _Traité de calcul diff. et de calcul int._ (3
tt., Paris, 1808-1819). There have been numerous later editions; a
translation by Herschel, Peacock and Babbage of an abbreviated edition
of Lacroix's treatise was published at Cambridge in 1816. G. Peacock,
_Examples of the Differential and Integral Calculus_ (Cambridge,
1820); A. L. Cauchy, _Résumé des leçons ... sur le calcul
infinitésimale_ (Paris, 1823), and _Leçons sur le calcul différentiel_
(Paris, 1829; _Oeuvres_, sér. 2, t. iv.); F. Minding, _Handbuch d.
Diff.-u. Int.-Rechnung_ (Berlin, 1836); F. Moigno, _Leçons sur le
calcul diff._ (4 tt., Paris, 1840-1861); A. de Morgan, _Diff. and Int.
Calc._ (London, 1842); D. Gregory, _Examples on the Diff. and Int.
Calc._ (2 vols., Cambridge, 1841-1846); I. Todhunter, _Treatise on the
Diff. Calc._ and _Treatise on the Int. Calc._ (London, 1852), numerous
later editions; B. Price, _Treatise on the Infinitesimal Calculus_ (2
vols., Oxford, 1854), numerous later editions; D. Bierens de Haan,
_Tables d'intégrales définies_ (Amsterdam, 1858); M. Stegemann,
_Grundriss d. Diff.- u. Int.-Rechnung_ (2 Bde., Hanover, 1862)
numerous later editions; J. Bertrand, _Traité de calc. diff. et int._
(2 tt., Paris, 1864-1870); J. A. Serret, _Cours de calc. diff. et
int._ (2 tt., Paris, 1868, 2nd ed., 1880, German edition by Harnack,
Leipzig, 1884-1886, later German editions by Bohlmann, 1896, and
Scheffers, 1906, incomplete); B. Williamson, _Treatise on the Diff.
Calc._ (Dublin, 1872), and _Treatise on the Int. Calc._ (Dublin, 1874)
numerous later editions of both; also the article "Infinitesimal
Calculus" in the 9th ed. of the _Ency. Brit._; C. Hermite, _Cours
d'analyse_ (Paris, 1873); O. Schlömilch, _Compendium d. höheren
Analysis_ (2 Bde., Leipzig, 1874) numerous later editions; J. Thomae,
_Einleitung in d. Theorie d. bestimmten Integrale_ (Halle, 1875); R.
Lipschitz, _Lehrbuch d. Analysis_ (2 Bde., Bonn, 1877, 1880); A.
Harnack, _Elemente d. Diff.- u. Int.-Rechnung_ (Leipzig, 1882, Eng.
trans. by Cathcart, London, 1891); M. Pasch, _Einleitung in d. Diff.-
u. Int.-Rechnung_ (Leipzig, 1882); Genocchi and Peano, _Calcolo
differenziale_ (Turin, 1884, German edition by Bohlmann and Schepp,
Leipzig, 1898, 1899); H. Laurent, _Traité d'analyse_ (7 tt., Paris,
1885-1891); J. Edwards, _Elementary Treatise on the Diff. Calc._
(London, 1886), several later editions; A. G. Greenhill, _Diff. and
Int. Calc._ (London, 1886, 2nd ed., 1891); É. Picard, _Traité
d'analyse_ (3 tt., Paris, 1891-1896); O. Stolz, _Grundzüge d. Diff.-
u. Int.-Rechnung_ (3 Bde., Leipzig, 1893-1899); C. Jordan, _Cours
d'analyse_ (3 tt., Paris, 1893-1896); L. Kronecker, _Vorlesungen ü. d.
Theorie d. einfachen u. vielfachen Integrale_ (Leipzig, 1894); J.
Perry, _The Calculus for Engineers_ (London, 1897); H. Lamb, _An
Elementary Course of Infinitesimal Calculus_ (Cambridge, 1897); G. A.
Gibson, _An Elementary Treatise on the Calculus_ (London, 1901); É.
Goursat, _Cours d'analyse mathématique_ (2 tt., Paris, 1902-1905);
C.-J. de la Vallée Poussin, _Cours d'analyse infinitésimale_ (2 tt.,
Louvain and Paris, 1903-1906); A. E. H. Love, _Elements of the Diff.
and Int. Calc._ (Cambridge, 1909); W. H. Young, _The Fundamental
Theorems of the Diff. Calc._ (Cambridge, 1910). A résumé of the
infinitesimal calculus is given in the articles "Diff.- u.
Int-Rechnung" by A. Voss, and "Bestimmte Integrale" by G. Brunel in
_Ency. d. math. Wiss._ (Bde. ii. A. 2, and ii. A. 3, Leipzig, 1899,
1900). Many questions of principle are discussed exhaustively by E. W.
Hobson, _The Theory of Functions of a Real Variable_ (Cambridge,
1907). (A. E. H. L.)
INFINITIVE, a form of the verb, properly a noun with verbal functions,
but usually taken as a mood (see GRAMMAR). The Latin grammarians gave it
the name of _infinitus_ or _infinitivus modus_, i.e. indefinite,
unlimited mood, as not having definite persons or numbers.
INFLEXION (from Lat. _inflectere_, to bend), the action of bending
inwards, or turning towards oneself, or the condition of being bent or
curved. In optics, the term "inflexion" was used by Newton for what is
now known as "diffraction of light" (q.v.). For inflexion in geometry
see CURVE. Inflexion when used of the voice, in speaking or singing,
indicates a change in tone, pitch or expression. In grammar (q.v.)
inflexion indicates the changes which a word undergoes to bring it into
correct relations with the other words with which it is used. In English
grammar nouns, pronouns, adjectives (in their degrees of comparison),
verbs and adverbs are inflected. Some grammarians, however, regard the
inflexions of adverbs more as an actual change in word-formation.
INFLUENCE (Late Lat. _influentia_, from _influere_, to flow in), a word
whose principal modern meaning is that of power, control or action
affecting others, exercised either covertly or without visible means or
direct physical agency. It is one of those numerous terms of astrology
(q.v.) which have established themselves in current language. From the
stars was supposed to flow an ethereal stream which affected the course
of events on the earth and the fortunes and characters of men. For the
law as to "undue influence" see CONTRACT.
INFLUENZA (syn. "grip," _la grippe_), a term applied to an infectious
febrile disorder due to a specific bacillus, characterized specially by
catarrh of the respiratory passages and alimentary canal, and occurring
mostly as an epidemic. The Italians in the 17th century ascribed it to
the influence of the stars, and hence the name "influenza." The French
name _grippe_ came into use in 1743, and those of _petite poste_ and
_petit courier_ in 1762, while _général_ became another synonym in 1780.
Apparently the scourge was common; in 1403 and 1557 the sittings of the
Paris law courts had to be suspended through it, and in 1427 sermons had
to be abandoned through the coughing and sneezing; in 1510 masses could
not be sung. Epidemics occurred in 1580, 1676, 1703, 1732 and 1737, and
their cessation was supposed to be connected with earthquakes and
volcanic eruptions.
The disease is referred to in the works of the ancient physicians, and
accurate descriptions of it have been given by medical writers during
the last three centuries. These various accounts agree substantially in
their narration of the phenomena and course of the disease, and
influenza has in all times been regarded as fulfilling all the
conditions of an epidemic in its sudden invasion, and rapid and
extensive spread. Among the chief epidemics were those of 1762, 1782,
1787, 1803, 1833, 1837 and 1847. It appeared in fleets at sea away from
all communication with land, and to such an extent as to disable them
temporarily for service. This happened in 1782 in the case of the
squadron of Admiral Richard Kempenfelt (1718-1782), which had to return
to England from the coast of France in consequence of influenza
attacking his crews.
Like cholera and plague, influenza reappeared in the last quarter of the
19th century, after an interval of many years, in epidemic or rather
pandemic form. After the year 1848, in which 7963 deaths were directly
attributed to influenza in England and Wales, the disease continued
prevalent until 1860, with distinct but minor epidemic exacerbations in
1851, 1855 and 1858; during the next decade the mortality dropped
rapidly though not steadily, and the diminution continued down to the
year 1889, In which only 55 deaths were ascribed to this cause. It is
not clear whether the disease ever disappears wholly, and the deaths
registered in 1889 are the lowest recorded in any year since the
registrar-general's returns began. Occasionally local outbreaks of
illness resembling epidemic influenza have been observed during the
period of abeyance, as in Norfolk in 1878 and in Yorkshire in 1887; but
whether such outbreaks and the so-called "sporadic" cases are
nosologically identical with epidemic influenza is open to doubt. The
relation seems rather to be similar to that between Asiatic cholera and
"cholera nostras." Individual cases may be indistinguishable, but as a
factor in the public health the difference between sporadic and epidemic
influenza is as great and unmistakable as that between the two forms of
cholera. This fact, which had been forgotten by some since 1847 and
never learnt by others, was brought home forcibly to all by the
visitation of 1889.
According to the exhaustive report drawn up by Dr H. Franklin Parsons
for the Local Government Board, the earliest appearances were observed
in May 1889, and three localities are mentioned as affected at the same
time, all widely separated from each other--namely, Bokhara in Central
Asia, Athabasca in the north-west Territories of Canada and Greenland.
About the middle of October it was reported at Tomsk in Siberia, and by
the end of the month at St Petersburg. During November Russia became
generally affected, and cases were noticed in Paris, Berlin, Vienna,
London and Jamaica (?). In December epidemic influenza became
established over the whole of Europe, along the Mediterranean, in Egypt
and over a large area in the United States. It appeared in several towns
in England, beginning with Portsmouth, but did not become generally
epidemic until the commencement of the new year. In London the full
onset of unmistakable influenza dated from the 1st of January 1890.
Everywhere it seems to have exhibited the same explosive character when
once fully established. In St Petersburg, out of a government staff of
260 men, 220 were taken ill in one night, the 15th of November. During
January 1890 the epidemic reached its height in London, and appeared in
a large number of towns throughout the British Islands, though it was
less prevalent in the north and north-west than in the south. January
witnessed a great extension of the disease in Germany, Holland,
Switzerland, Austria-Hungary, Italy, Spain and Portugal; but in Russia,
Scandinavia and France it was already declining. The period of greatest
activity in Europe was the latter half of December and the earlier half
of January, with the change of the year for a central point. Other parts
of the world affected in January 1890 were Cape Town, Canada, the United
States generally, Algiers, Tunis, Cairo, Corsica, Sardinia, Sicily,
Honolulu, Mexico, the West Indies and Montevideo. In February the
provincial towns of England were most severely affected, the death-rate
rising to 27.4, but in London it fell from 28.1 to 21.2, and for Europe
generally the back of the epidemic was broken. At the same time,
however, it appeared in Ceylon, Penang, Japan, Hong Kong and India; also
in West Africa, attacking Sierra Leone, and Gambia in the middle of the
month; and finally in the west, where Newfoundland and Buenos Aires were
invaded. In March influenza became widely epidemic in India,
particularly in Bengal and Bombay, and made its appearance in Australia
and New Zealand. In April and May it was epidemic all over Australasia,
in Central America, Brazil, Peru, Arabia and Burma. During the summer
and autumn it reached a number of isolated islands, such as Iceland, St
Helena, Mauritius and Réunion. Towards the close of the year it was
reported from Yunnan in the interior of China, from the Shiré Highlands
in Central Africa, Shoa in Abyssinia, and Gilgit in Kashmir. In the
course of fifteen months, beginning with its undoubted appearance in
Siberia in October 1889, it had traversed the entire globe.
The localities attacked by influenza in 1889-1890 appear in no case to
have suffered severely for more than a month or six weeks. Thus in
Europe and North America generally the visitation had come to an end in
the first quarter of 1890. The earliest signs of an epidemic revival on
a large scale occurred in March 1891, in the United States and the north
of England. It was reported from Chicago and other large towns in the
central states, whence it spread eastwards, reaching New York about the
end of March. In England it began in the Yorkshire towns, particularly
in Hull, and also independently in South Wales. In London influenza
became epidemic for the second time about the end of April, and soon
afterwards was widely distributed in England and Wales. The large towns
in the north, together with London and Wales, suffered much more heavily
in mortality than in the previous attack, but the south-west of England,
Scotland and Ireland escaped with comparatively little sickness. The
same may be said of the European continent generally, except parts of
Russia, Scandinavia and perhaps the north of Germany. This second
epidemic coincided with the spring and early summer; it had subsided in
London by the end of June. The experience of Sheffield is interesting.
In 1890 the attack, contrary to general experience, had been undecided,
lingering and mild; in 1891 it was very sudden and extremely severe, the
death-rate rising to 73.4 during the month of April, and subsiding with
equal rapidity. During the third quarter of the year, while Europe was
free, the antipodes had their second attack, which was more severe than
the first. As in England, it reversed the previous order of things,
beginning in the provinces and spreading thence to the capital towns.
The last quarter of the year was signalized by another recrudescence in
Europe, which reached its height during the winter. All parts, including
Great Britain, were severely affected. In England those parts which had
borne the brunt of the epidemic in the early part of the year escaped.
In fact, these two revivals may be regarded as one, temporarily
interrupted by the summer quarter.
The recrudescence at the end of 1891 lasted through mid-winter, and in
many places, notably in London, it only reached its height in January
1892, subsiding slowly and irregularly in February and March. Brighton
suffered with exceptional severity. The continent of Europe seems to
have been similarly affected. In Italy the notifications of influenza
were as follow: 1891--January to October, 0; November, 30; December,
6461; 1892--January, 84,543; February, 55,352; March, 28,046; April,
7962; May, 1468; June, 223. Other parts of the world affected were the
West Indies, Tunis, Egypt, Sudan, Cape Town Teheran, Tongking and China.
In August 1892 influenza was reported from Peru, and later in the year
from various places in Europe.
A fourth recrudescence, but of a milder character, occurred in Great
Britain in the spring of 1893, and a fifth in the following winter, but
the year 1894 was freer from influenza than any since 1890. In 1895
another extensive epidemic took place. In 1896 influenza seemed to have
spent its strength, but there was an increased prevalence of the disease
in 1897, which was repeated on a larger scale in 1898, and again in
1899, when 12,417 deaths were recorded in England and Wales. This was
the highest death-rate since 1892. After this the death-rate declined to
half that amount and remained there with the slight upward variations
until 1907, in which the total death-rate was 9257. The experience of
other countries has been very similar; they have all been subjected to
periodical revivals of epidemic influenza at irregular intervals and of
varying intensity since its reappearance in 1889, but there has been a
general though not a steady decline in its activity and potency. Its
behaviour is, in short, quite in keeping with the experience of
1847-1860, though the later visitation appears to have been more violent
and more fatal than the former. Its diffusion was also more rapid and
probably more extensive.
The foregoing general summary may be supplemented by some further
details of the incidence in Great Britain. The number of deaths directly
attributed to influenza, and the death-rates per million in each year in
England and Wales, are as follow:--
+------+--------+-------------+
| Year.| Deaths.| Death-rates |
| | | per million.|
+------+--------+-------------+
| 1890 | 4,523 | 157 |
| 1891 | 16,686 | 574 |
| 1892 | 15,737 | 534 |
| 1893 | 9,669 | 325 |
| 1894 | 6,625 | 220 |
| 1895 | 12,880 | 424 |
| 1896 | 3,753 | 122 |
| 1897 | 6,088 | 196 |
| 1898 | 10,405 | 331 |
| 1899 | 12,417 | 389 |
| 1900 | 16,245 | 504 |
| 1901 | 5,666 | 174 |
| 1902 | 7,366 | 223 |
| 1903 | 6,322 | 189 |
| 1904 | 5,694 | 168 |
| 1905 | 6,953 | 204 |
| 1906 | 6,310 | 183 |
| 1907 | 9,257 | 265 |
+------+--------+-------------+
It is interesting to compare these figures with the corresponding ones
for the previous visitation:--
+------+--------+-------------+
| Year.| Deaths.| Death-rates |
| | | per million.|
+------+--------+-------------+
| 1847 | 4,881 | 285 |
| 1848 | 7,963 | 460 |
| 1849 | 1,611 | 92 |
| 1850 | 1,380 | 78 |
| 1851 | 2,152 | 120 |
| 1852 | 1,359 | 76 |
| 1853 | 1,789 | 99 |
| 1854 | 1,061 | 58 |
| 1855 | 3,568 | 193 |
+------+--------+-------------+
The two sets of figures are not strictly comparable, because, during the
first period, notification of the cause of death was not compulsory; but
it seems clear that the later wave was much the more deadly. The average
annual death-rate for the nine years is 320 in the one case against 162
in the other, or as nearly as possible double. In both epidemic periods
the second year was far more fatal than the first, and in both a marked
revival took place in the ninth year; in both also an intermediate
recrudescence occurred, in the fifth year in one case, in the sixth in
the other. The chief point of difference is the sudden and marked drop
in 1849-1850, against a persistent high mortality in 1892-1893,
especially in 1892, which was nearly as fatal as 1891.
To make the significance of these epidemic figures clear, it should be
added that in the intervening period 1861-1889 the average annual
death-rate from influenza was only fifteen, and in the ten years
immediately preceding the 1890 outbreak it was only three. Moreover, in
epidemic influenza, the mortality directly attributed to that disease is
only a fraction of that actually caused by it. For instance, in January
1890 the deaths from influenza in London were 304, while the excess of
deaths from respiratory diseases was 1454 and from all causes 1958 above
the average.
We have seen above that the mortality was far greater in the second
epidemic year than in the first, and this applies to all parts of
England, and to rural as well as to urban communities, as the following
table shows:--
_Deaths from Influenza._
+--------------------------------------+------+------+
| | 1890.| 1891.|
+--------------------------------------+------+------+
| London | 624 | 2302 |
| 24 Great Towns over 80,000 population| 439 | 2417 |
| 35 Towns between 20,000 and 80,000 | 186 | 765 |
| 21 Towns between 10,000 and 20,000 | 46 | 196 |
| 60 Towns under 10,000 | 62 | 196 |
| 85 Rural Sanitary Districts | 317 | 841 |
+--------------------------------------+------+------+
In spite of these figures, it appears that the 1890 attack, which was in
general much more sudden in its onset than that of 1891, also caused a
great deal more sickness. More people were "down with influenza," though
fewer died. For Instance, the number of persons treated at the Middlesex
Hospital in the two months' winter epidemic of 1890 was 1279; in the far
more fatal three months' spring epidemic of 1891 it was only 726. One
explanation of this discrepancy between the incidence of sickness and
mortality is that in the second attack, which was more protracted and
more insidious, the stress of the disease fell more upon the lungs.
Another is that its comparative mildness, combined with the time of
year, in itself proved dangerous, because it tempted people to disregard
the illness, whereas in the first epidemic they were too ill to resist.
On the whole, rural districts showed a higher death-rate than towns, and
small towns a higher one than large ones in both years. This is
explained by the age distribution in such localities; influenza being
particularly fatal to aged people, though no age is exempt. Certain
counties were much more severely affected than others. The eastern
counties, namely, Essex, Suffolk and Norfolk, together with Hampshire
and one or two others, escaped lightly in both years; the western
counties, namely, North and South Wales, with the adjoining counties of
Monmouth, Hereford and Shropshire, suffered heavily in both years.
It will be convenient to discuss _seriatim_ the various points of
interest on which light has been thrown by the experience described
above.
The bacteriology of influenza is discussed in the article on PARASITIC
DISEASES. The disease is often called "Russian" influenza, and its
origin in 1889 suggests that the name may have some foundation in fact.
A writer, who saw the epidemic break out in Bokhara, is quoted by him to
the following effect:--"The summer of 1888 was exceptionally hot and
dry, and was followed by a bitterly cold winter and a rainy spring. The
dried-up earth was full of cracks and holes from drought and subsequent
frost, so that the spring rains formed ponds in these holes, inundated
the new railway cuttings, and turned the country into a perfect marsh.
When the hot weather set in the water gave off poisonous exhalations,
rendering malaria general." On account of the severe winter, the people
were enfeebled from lack of nourishment, and when influenza broke out
suddenly they died in large numbers. Europeans were very severely
affected. Russians, hurrying home, carried the disease westwards, and
caravans passing eastwards took it into Siberia. There is a striking
similarity in the conditions described to those observed in connexion
with outbreaks of other diseases, particularly typhoid fever and
diphtheria, which have occurred on the supervention of heavy rain after
a dry period, causing cracks and fissures in the earth. Assuming the
existence of a living poison in the ground, we can easily understand
that under certain conditions, such as an exceptionally dry season, it
may develop exceptional properties and then be driven out by the
subsequent rains, causing a violent outbreak of illness. Some such
explanation is required to account for the periodical occurrence of
epidemic and pandemic diffusions starting from an endemic centre. We may
suppose that a micro-organism of peculiar robustness and virulence is
bred and brought into activity by a combination of favourable
conditions, and is then disseminated more or less widely according to
its "staying power," by human agency. Whether central Asia is an endemic
centre for influenza or not there is no evidence, but the disease seems
to be more often prevalent in the Russian Empire than elsewhere.
Extensive outbreaks occurred there in 1886 and 1887, and it is certain
that the 1889 wave was active in Siberia at an earlier date than in
Europe, and that it moved eastwards. The hypothesis that it originated
in China is unsupported by evidence. But whatever may be the truth with
regard to origin, the dissemination of influenza by human agency must be
held to be proved. This is the most important addition to our knowledge
of the subject contributed by recent research. The upshot of the inquiry
by Dr Parsons was to negative all theories of atmospheric influence, and
to establish the conclusion that the disease was "propagated mainly,
perhaps entirely, by human intercourse."
He found that it prevailed independently of climate, season and
weather; that it moved in a contrary direction to the prevailing
winds; that it travelled along the lines of human intercourse, and not
faster than human beings can travel; that in 1889 it travelled much
faster than in previous epidemics, when the means of locomotion were
very inferior; that it appeared first in capital towns, seaports and
frontier towns, and only affected country districts later; that it
never commenced suddenly with a large number of cases in a place
previously free from disease, but that epidemic manifestations were
generally preceded for some days or weeks by scattered cases; that
conveyance of infection by individuals and its introduction into fresh
places had been observed in many instances; that persons brought much
into contact with others were generally the first to suffer; that
persons brought together in large numbers in enclosed spaces suffered
more in proportion than others, and that the rapidity and extent of
the outbreak in institutions corresponded with the massing together of
the inmates.
These conclusions, based upon the 1889-1890 epidemic, have been
confirmed by subsequent experience, especially in regard to the complete
independence of season and weather shown by influenza. It has appeared
and disappeared at all seasons and in all weathers and only popular
ignorance continues to ascribe its behaviour to atmospheric conditions.
In Europe, however, it has prevailed more often in winter than in
summer, which may be due to the greater susceptibility of persons in
winter, or, more probably, to the fact that they congregate more in
buildings and are less in the open air during that part of the year. No
doubt is any longer entertained of its infectious character, though the
degree of infectivity appears to vary considerably. Many cases have been
recorded of individuals introducing it into houses, and of all or most
of the other inmates then taking it from the first case. Difficulties in
preventing the spread of infection are due to (1) the shortness of the
period of incubation, (2) the disease being infectious in the earliest
stages before the nature of the illness is recognized, (3) the milder
varieties being equally infectious with the severe attacks, and the
patient going to work and spreading the infection, (4) the diagnosis
often being difficult, influenza being possibly confused with ordinary
catarrhal attacks, typhoid fever and other diseases. Domestic animals
seem to be free from any suspicion of being liable to human influenza.
Sanitary conditions, other than overcrowding, do not appear to exercise
any influence on the spread of influenza.
Influenza has been shown to be an acute specific fever having nothing
whatever to do with a "bad cold." There may be some inflammation of the
respiratory passages, and then symptoms of catarrh are present, but that
is not necessarily the case, and in some epidemics such symptoms are
quite exceptional. This had been recognized by various writers before
the 1889 visitation, but it had not been generally realized, as it has
been since, and some medical authorities, who persisted in regarding
influenza as essentially a "catarrhal" affection, were chiefly to blame
for a widespread and tenacious popular fallacy.
Leichtenstern, in his masterly article in Nothnagel's _Handbuch_,
divides the disease as follows:--(1) Epidemic influenza vera caused by
Pfeiffer's bacillus; (2) Endemic-epidemic influenza vera, which occurs
several years after a pandemic and is caused by the same bacillus; (3)
Endemic influenza nostras or eatarrhal fever, called _la grippe_, and
bearing the same relation to true influenza as cholera nostras does to
Asiatic cholera.
The "period of incubation" is one to four days. Susceptibility varies
greatly, but the conditions that influence it are matters of conjecture
only. It appears that the inhabitants of Great Britain are less
susceptible than those of many other countries. Dr Parsons gives the
following list, showing the proportion of the population estimated to
have been attacked in the 1889-1890 epidemic in different localities:--
+---------------------+---------+---------------------+---------+
| Place. |Per cent.| Place. |Per cent.|
+---------------------+---------+---------------------+---------+
| St Petersburg | 50 | Portugal | 90 |
| Berlin | 33 | Vienna | 30-40 |
| Nuremberg | 67 | Belgrade | 33 |
| Grand-Duchy of Hesse| 25-30 | Antwerp | 33 |
| Grand-Duchy, other | | Gaeta | 50-77 |
| Districts | 50-75 | Massachusetts | 39 |
| Heligoland | 50 | Peking | 50 |
| Budapest | 50 | St Louis (Mauritius)| 67 |
+---------------------+---------+---------------------+---------+
In and about London he reckoned roughly from a number of returns that
the proportion was about 12½% among those employed out of doors and
25% among those in offices, &c. The proportion among the troops in the
Home District was 9.3%. The General Post Office made the highest return
with 33.6%, which is accounted for partly by the enormous number of
persons massed together in the same room in more than one department,
and partly by the facilities for obtaining medical advice, which would
tend to bring very light cases, unnoticed elsewhere, upon the record. No
public service was seriously disorganized in England by sickness in the
same manner as on the continent of Europe. Some individuals appear to be
totally immune; others take the disease over and over again, deriving no
immunity, but apparently greater susceptibility from previous attacks.
The symptoms were thus described by Dr Bruce Low from observations made
in St Thomas's Hospital, London, in January 1890:--
The invasion is sudden; the patients can generally tell the time when
they developed the disease; e.g. acute pains in the back and loins
came on quite suddenly while they were at work or walking in the
street, or in the case of a medical student, while playing cards,
rendering him unable to continue the game. A workman wheeling a barrow
had to put it down and leave it; and an omnibus driver was unable to
pull up his horses. This sudden onset is often accompanied by vertigo
and nausea, and sometimes actual vomiting of bilious matter. There are
pains in the limbs and general sense of aching all over; frontal
headache of special severity; pains in the eyeballs, increased by the
slightest movement of the eyes; shivering; general feeling of misery
and weakness, and great depression of spirits, many patients, both men
and women, giving way to weeping; nervous restlessness; inability to
sleep, and occasionally delirium. In some cases catarrhal symptoms
develop, such as running at the eyes, which are sometimes injected on
the second day; sneezing and sore throat; and epistaxis, swelling of
the parotid and submaxillary glands, tonsilitis, and spitting of
bright blood from the pharynx may occur. There is a hard, dry cough of
a paroxysmal kind, worst at night. There is often tenderness of the
spleen, which is almost always found enlarged, and this persists after
the acute symptoms have passed. The temperature is high at the onset
of the disease. In the first twenty-four hours its range is from 100°
F. in mild cases to 105° in severe cases.
Dr J. S. Bristowe gave the following description of the illness during
the same epidemic:--
The chief symptoms of influenza are, coldness along the back, with
shivering, which may continue off and on for two or three days; severe
pain in the head and eyes, often with tenderness in the eyes and pain
in moving them; pains in the ears; pains in the small of the back;
pains in the limbs, for the most part in the fleshy portions, but also
in the bones and joints, and even in the fingers and toes; and febrile
temperature, which may in the early period rise to 104° or 105° F. At
the same time the patient feels excessively ill and prostrate, is apt
to suffer from nausea or sickness and diarrhoea, and is for the most
part restless, though often (and especially in the case of children
and those advanced in age) drowsy.... In ordinary mild cases the above
symptoms are the only important ones which present themselves, and the
patient may recover in the course of three or four days. He may even
have it so mildly that, although feeling very ill, he is able to go
about his ordinary work. In some cases the patients have additionally
some dryness or soreness of the throat, or some stiffness and
discharge from the nose, which may be accompanied by slight bleeding.
And in some cases, for the most part in the course of a few days, and
at a time when the patient seems to be convalescent, he begins to
suffer from wheezing in the chest, cough, and perhaps a little
shortness of breath, and before long spits mucus in which are
contained pellets streaked or tinged with blood.... Another
complication is diarrhoea. Another is a roseolous spotty rash....
Influenza is by no means necessarily attended with the catarrhal
symptoms which the general public have been taught to regard as its
distinctive signs, and in a very large proportion of cases no
catarrhal condition whatever becomes developed at any time.
Several writers have distinguished four main varieties of the
disease--namely, (1) nervous, (2)gastro-intestinal, (3)respiratory, (4)
febrile, a form chiefly found in children. Clifford Allbutt says,
"Influenza simulates other diseases." Many forms are of typhoid or
comatose types. Cardiac attacks are common, not from organic disease but
from the direct poisoning of the heart muscle by influenza.
Perhaps the most marked feature of influenza, and certainly the one
which victims have learned to dread most, is the prolonged debility and
nervous depression that frequently follow an attack. It was remarked by
Nothnagel that "Influenza produces a specific nervous toxin which by its
action on the cortex produces psychoses." In the Paris epidemic of 1890
the suicides increased 25%, a large proportion of the excess being
attributed to nervous prostration caused by the disease. Dr Rawes,
medical superintendent of St Luke's hospital, says that of insanities
traceable to influenza melancholia is twice as frequent as all other
forms of insanity put together. Other common after-effects are
neuralgia, dyspepsia, insomnia, weakness or loss of the special senses,
particularly taste and smell, abdominal pains, sore throat, rheumatism
and muscular weakness. The feature most dangerous to life is the special
liability of patients to inflammation of the lungs. This affection must
be regarded as a complication rather than an integral part of the
illness. The following diagram gives the annual death-rate per million
in England and Wales, and is taken from an article by Dr Arthur
Newsholme in _The Practitioner_ (January 1907).
The deaths directly attributed to influenza are few in proportion to the
number of cases. In the milder forms it offers hardly any danger to life
if reasonable care be taken, but in the severer forms it is a fairly
fatal disease. In eight London hospitals the case-mortality among
in-patients in the 1890 outbreak was 34.5 per 1000; among all patients
treated it was 1.6 per 1000. In the army it was rather less.
The infectious character of influenza having been determined,
suggestions were made for its administrative control on the familiar
lines of notification, isolation and disinfection, but this has not
hitherto been found practicable. In March 1895, however, the Local
Government Board issued a memorandum recommending the adoption of the
following precautions wherever they can be carried out:--
1. The sick should be separated from the healthy. This is especially
important in the case of first attacks in a locality or a household.
2. The sputa of the sick should, especially in the acute stage of the
disease, be received into vessels containing disinfectants. Infected
articles and rooms should be cleansed and disinfected.
3. When influenza threatens, unnecessary assemblages of persons should
be avoided.
4. Buildings and rooms in which many people necessarily congregate
should be efficiently aerated and cleansed during the intervals of
occupation.
There is no routine treatment for influenza except bed. In all cases bed
is advisable, because of the danger of lung complications, and in mild
ones it is sufficient. Severer ones must be treated according to the
symptoms. Quinine has been much used. Modern "anti-pyretic" drugs have
also been extensively employed, and when applied with discretion they
may be useful, but patients are not advised to prescribe them for
themselves.
[Illustration]
Sir Wm. Broadbent in a note on the prophylaxis of influenza recommends
quinine in a dose of two grains every morning, and remarks: "I have had
opportunities of obtaining extraordinary evidence of its protective
power. In a large public school it was ordered to be taken every
morning. Some of the boys in the school were home boarders, and it was
found that while the boarders at the school took the quinine in the
presence of a master every morning, there were scarcely any cases of
influenza among them, although the home boarders suffered nearly as much
as before." He continues, "In a large girls' school near London the same
thing was ordered, and the girls and mistresses took their morning dose
but the servants were forgotten. The result was that scarcely any girl
or mistress suffered while the servants were all down with influenza."
The liability to contract influenza, and the danger of an attack if
contracted, are increased by depressing conditions, such as exposure to
cold and to fatigue, whether mental or physical. Attention should,
therefore, be paid to all measures tending to the maintenance of health.
Persons who are attacked by influenza should at once seek rest, warmth
and medical treatment, and they should bear in mind that the risk of
relapse, with serious complications, constitutes a chief danger of the
disease.
In addition to the ordinary text-books, see the series of articles by
experts on different aspects in _The Practitioner_ (London) for
January 1907.