f yields a result definite only as to form, up to an additive constant, C. (Cf. fč

Evolute and Involute, above). Derivation simplifies, reducing even transcendents to algebraics; Integration complicates, lifting algebraics and even rationals up into tran4. Derivation is deductive and can create no new forms. Integration is inductive and creates an ∞ of new forms, all defined as integrals.

scendents (as √a2-x2 and

x

2

cos x cos x dx=sin x cos x +

=sin x cos x + x·

2

Jsi

2

sinx.dr

cos xdx; whence

cos x .dx=x+sin x cos x

Similarly for

any integral power of sine or cosine except (sin x)-1, (cos x)-', which are reduced by passing to the half-angle, .

x

What is the range of such reductions? What functions can we thus integrate in terms of known functions? Few enough. Of Algebraics, (x) $(x)'

I. Rational functions, tion into part-fractions.

by decomposi

II. Rational functions of x and (ax+1)a/B

Operating directly on y=f(x) by a series of differentiations as to x, say (D), we get some function of x, as X, or (D)y=X. If we know X and, we may seek that y=f(x) that will yield X on being subjected to the train of operations (D); i.e. we seek to invert at once the totality of operations (D), X so that y =-(D)X. This inversion $(D) is solving the Differential Equation (D)y=X, and is perhaps the most profound of mathematical operations, of immense and even unconquerable difficulty, overcome as yet only in day dy special cases. Thus x2 +4y=2x2, If we think of y2=ax2+2bx+c as a conic, and y-ẞ u(x-α) as a secant through a where &(D) =x2D2 −3xD+4, yields, as result point (a, 3) of the conic, then we may express

y=x2(A+B log x+log x2), where A and B are arbitrary constants. Other forms of 4(D), quite as simple, yield far higher transcendents.

Inverting a table of elementary D's we get a table of elementary Integrals. The art is to reduce other forms, if possible, to these elementary forms; when impossible, we must introduce transcendents defined by integrals.

Change of Variable is the most fruitful method of Reduction. By mediate derivation, Dip(u) = '(u)• ux; also Dup(u) = '(u);

L.C.M. of B, B', ... Herewith the
rational in u.

III. Rational functions of x and
√ax2 +20x+c.

both x and y rationally through u, which reduces this case to I; (a, B) may be taken variously. Generally we bring y2 to the form of sum or difference of two squares by putting x=z+h.

IV. Rational functions of x and y, these being coördinates of a unicursal F(x, y) =o. We shall then have x=(u), y=4(u), where and are rational, whereby these Abelian Integrals reduce to I.

The binomial x "(a+bxn)p can be reduced to II, and hence integrated, not generally, but in these important cases, by putting u=x" 1. integral; if (r and s integers),

hences p'(u)du = p(u) = f(u)-uz-dx. In this put =z.
f, all is supposed expressed through x.
20 S,
Hence, to pass from an old to a new variable put a + bu=2.

of integration, multiply by the D of the old
as to the new, or divide by the D of the new
as to the old; 2.e. under the sign,

n

=

(rands integers),

3. +p integral; if p=(r and s integers), put a+bu=uz3.

If f(x) be rational in x, Vax+b, Vcx+d, put ax+b=v2, which reduces it to 3. Of Transcendentals.—1. Rational functions of sin x and cos x. Put u=1 =tan a very impor tant substitution.

χ

2

2. Rational functions of ear ̧ Put uea¤ ̧ 3. Rational Integral functions of x, eax ebx,... sin mx, sin nx, cos mx, cos nx, Express the sines and cosines through imagiexpress the Here imaginaries through sines and cosines.

· Su,vdx+fuvzdx, or fuv.dx-uv-fuvdx. nary exponentials.

In the result

CALCULUS, THE INFINITESIMAL

1 = L £j(xk) dk = [] f(x)dx exists when f(x) = f(x, y), and the cylindric surface standing

throughout [a, b]; but also (2) when f(x) is V=L2j(x,y)4R = ƒ f(x

finitely discontinuous at a finite number of

CALCULUS, THE INFINITESIMAL

must be indifferent in what order the elements

From DI we readily pass to the triple :,

4R are taken; hence we may sum frat such a fi(x, y, z)dx dy dz, and hence to mul

a strip parallel to X, and then sum all
strips along Y. This double summing is ex-
pressed by a Double Integral (II) thus:

Here for any value of x the values of y are determined by the equation of B. Hence b and are functions of x; but a and a depend on the extreme parallels to Y tangent to B, hence are absolute constants.

a

tiple I's in general. The higher Jacobian maintains its rôle: du do dw = ] dx dy dz, J =

Vx Vy Vz ; and so on in general. Thus, to pass from rectangular to polar coordinates, x =pcos &, y=p sind; J(x,y; p, 0) =p; under ff, dx dy = pdød &—this latter is in

curvilinear volumetric element under Analogy readily extends these forms to n-fold spaces. Thus the Jacobian appears geometrically as a real derivative, the limit of the ratio of two simultaneous changes.

It is geometrically clear that II is perfectly fact the elementary curvilinear rectangle.-To definite, but we must ask in default of Geometry, pass from rectangular to spherical coordinates, when does approach the same limit index= cos sin o, y=p sin sin o, -p cos &, pendently of the function-value chosen for each whence J(x, y, z; p,, ) = p2 sin p, and AR and of the way in which each 4Ro, as their 2 sin pdp dedo is in fact the rectangular number∞o? Answer: When ΣDk 4Kko as each 4R=0, Dk being the greatest fluctuation in function-value in Rk. When is this the the case? Answer: (1) When f(x, y) is continuous throughout R; (2) when f at single points or on single lines (at points) becomes finitely discontinuous or indeterminate or oscillatory; (3) when f becomes thus finitely discontinuous or indeterminate oscillatory along an of lines (at 2 points), if only the sum of the elements (4R's), where D>o, is itself < (infinitesimal); i.e. when the

or

a

•b

The single ¡(x)dx=F(b)− F(a), where f=F' expresses the sum-value of f, integrated along the length b-a, through the end-values of

linear masses do not form an areal (r planar) some Fat banda; can the double ƒ ƒ j(x, y) dx dy

mass, i.e. when their initial elements form not a linear but only a discrete mass.

May f(x, y) attain and II retain sense? Answer: Iff attains a definite, but only at definite points, or along a curve and of

integrated over the region R also be expressed through the end-values of some F(x, y) along the contour of R? This query is much harder to answer, but is answered similarly: If f(x, y)

order <1, then the II remains definite and is integrable in R, then in general f*¡(x, y)dx

finite; also the order of integration remains indifferent. Here the contribution, to the II, o as the element of area (in XY) shrinks toward o along the curve; i.e. the volume V

is for every included value of y a continuous function of x, F(x, y), and for every included value of x an integrable function of y.

Then

shoots up to only along an infinitely sharp theDI ffi(x, y)dx dy=fF(x, y) sin yds, where

s is the contour of R and slope of the normal (drawn inwards at any point of s) to the

+ Y-axis. This latter is a curvilinear S,

geometrically depicted as a wall built up

Extremely important is the change of vari- (resp. down) along the contour s of R.—

ables in II. In simple integration du f, whereby we pass from x

under the

uda Similarly a fff of f extended throughout

to

u

a volume (or three-wayed spread) may be as variable of integration. In passing from expressed through the end-values of a certain F integrated over the entire surface (S) of a space-integral is turned into a surface-integral; and conversely:

მ მა au av U x V x дх ду ду дх [Uy Vy] (u, v) =J(u, v; x, y). This remarkable d(x, y) expression, introduced by Jacobi and named by him Functional Determinant, is called Jacobian (Salmon). As already exemplified, it plays the rôle of derivative of the system ə(u, v) ə(x, y) (u, v) as to (x, y). In fact, (x, y) (u, v) ə (u, v) ở (w, z) (w, z) "d(x, y)'

VOL. 4-7

CALCULUS, THE INFINITESIMAL

common limit of the length of inscribed and circumscribed polygons of which each sideo.

the integrand first, and then integrate, we
Hence the order
obtain the same result.
Integr. Der.p is equivalent to Der.p Integra.
af(x. P) is in general

Since st=√(x1)2+(yı)2, s=f√(xi)2+(yı)2dt. This holds generally, if for a definite interval

not

If now x=4(t), y=4(t), be continuous
functions of t, with finite limits of value,
then this integration is possible, the curve
the
Such is
case in
is rectifiable.
Weierstrass's curve; there the oscillations
(maxima and minima) are infinitely many in
every neighborhood however small, nor is the
variation finite in any such neighborhood-
the arc is infinite between any two points.

Volume is given by triple Integration,

SSS dx dy dz, extremes defined by the bounding

surface. Often the area of a section perpendicular (or possibly oblique) to an axis, as X,

=

[p, p+4p], and for [a, b],

др

(with possible exception of only discrete masses of points) a continuous function of both x and p; then

მ

3 fi(x,
f(x, p)dx =
dx = · fajx, Pdx.

др

If p appears in either a or b or both, then

a

apf f(x, p)dx

be an integrable function of p, F(p), in [a, B], integrating as top we get

is a function of x, „S=j(x); then V-Sdx. f(x, p)dx' dp-f" F(p)dp, and the

In the important case of Revolutes (of an area bounded by X, the curve, and two y-ordinates),

zS =ny2, V=nfy2dx.

Quadrature of a curved surface is sometimes called Complanation.

Here again the area must be defined as the common limit of the surface area of polyhedra inscribed and circumscribed, no The surface element dS or (45) matter how. about P may be viewed as projected into the element dx dy in XY and as having a limiting ratio 1 with the corresponding element 4II in the plane tangent at P. The slope of this plane to XY=7=slope of normal to Z; AS 48 ΔΠ hence Lim. Ax Ay JI JxJy sec r={Fx2 + F y2 + F z2 } }/Fz;

Lim.

Among such the Eulerians are conspicuous, especially that of 2d species (Legendre),

fx-le-dx, denoted by ľ(a), or better by

the Gaussian II(a-1), through which countless others are expressible. This Ã- or II-function has remarkable properties:

ff/Fdx dy, the region of integration in the factonal property, n=n\n=1.

XY being the projection thereon of S, under
obvious conditions. Often the surface is
given parametrically, i.e., x, y, and z as func-
Then we
tions of the independents, u and v.
Xu Yu Zu ||
write off the rectangular array Xv Yv Zv il
and form therefrom the three Jacobians, by
deleting the columns in order, commonly
written A, B, C. Hence, by easy substitution
for F1, Fy, F., S-VA2+B2+C2 du dv.*
To every point (x, y, z) of S there is coordinated
a point (uv) of a plane.

For Revolutes, S=27 yds.

CALCULUS, THE INFINITESIMAL

a

Cauchy's, Schlömilch's, come at once on apply. This ing the Maximum-Minimum Theorem. development holds under the two necessary and sufficient conditions (Pringsheim):

1. That p(x) possess everywhere in [x, x+R} definite finite differential coefficients of every finite order;

2. That Lim. · f(n) (xo+h)•kn converge uni().formly on o (for no) for all pairs (h, k) · for which oh≤h+k<R.

(1+x)a+b

dx,

is denoted by B(a, b) (Binet) and is connected
with the 2d by B(a, b)I(a+b) =Ï'(a)⋅Ib, and
being expressible thus simply through has
not so much independent significance.
The central notion of the Integral Calculus,
the LIMIT of a SUM, is more obvious than that
of the Differential Calculus, the LIMIT of a
QUOTIENT. The foundations of the one are
also seen to be much broader than those of the
other, so that the former is not merely the inverse
of the latter. The twain seize upon the two
great aspects of History, the Dynamical and
the Statical, Process and Result. While the
Integral Calculus borrows its speed and direct-
ness from the Differential Calculus, its own
reaction upon this latter is instant and power-
ful. Thus, from integration by parts we have

The Infinitesimal Analysis or Method of Limits is very highly developed and is applicable to almost every subject of exact thought, often asserting itself in the most surprising fashion, as in the Theories of Numbers and of Knots, to which it might seem wholly alien, suddenly unlocking and laying wide open secret .passages utterly unsuspected. In particular the Integral Calculus shows itself amazingly and unendingly fertile in the generation of new notions. As other and still other fields are exposed to investigative thought, the Calculus will receive more and more applications, and there seems to be no limit to the subtlety and refinement of its processes, to the keenness and penetration that may be given to this twoedged sword of the spirit, the strongest, sharpest, and most flexible ever fashioned or wielded by the mind of man.

Historical Sketch.-Passing by anticipations, especially of Integration, that reach back at to Barrow's Lectiones opticæ et geometrica least to Archimedes (287-212 B.C.), we come (1669-70), on which Newton collaborated, how

much no one knows. Barrow used the
Differential Triangle even in 1664 (indefinite
parvum, . . . ob indefinitam curvæ parvitu
dinem), ca ling dy a and 4x e (as Fermat used
A and E, 1638). Newton was busied with
Series at Cambridge, 1665-6 (eo tempore pestis
ingruens coegit me hinc fagere-in his famous
letter, filling thirty pages in the Opuscula,
24th October, 1676).
to Oldenburg, Secretary of the Royal Society,
His MS. De analysi
per æquationes numero terminorum infinitás

ƒ $′(y)dy=y&′(y) — ƒ 4′′ (yydy; -"(y)ydy (partially published first in Wallis's Works,

vol. II, 1693) was shown to Barrow, Collins, Lord Brouncker in 1669, wherein he used o for a magnitude ultimately vanishing, as had James Gregory already in his (Geometriæ pars universalis (1667, Venice). Remainder); He treated Rectification, Cubature, and Mass-Center determinations as reducible to Quadrature and to be solved by introducing the notion of (Momentum)=instantaneous change, thus going beyond Barrow. Newton's Methodus Fluxionum et Serierum infinitarum' was

42 $(h) = p(0)+hp′(h) - - p'' (h) + —p''' (h) — ... 12 3 ́ynq(n+1)(y)dy. To avoid the alterna

tion in sign we take | p(a−u)du and proceed as ready for the press before 1672, but not printed

before: then on putting a=x,+h,

4(x+h) = f(x)+hp'(xo)+

+

I

till 1736. In it he proposes, (1) to find the velocity at any instant from the space traversed up to each instant, (2) to find the latter (space) from the former (velocity)-the two problems of Derivation and Integration conceived kinematically. The equicrescent mag

funqn+1(x+h−u)du, o<h<R. nitude x, as a space, is called fluens (Cavalieri

Such is the swiftest, directest, nearest-lying deduction of the fundamental Taylor's Series, by which the value of at (x+h) is built up out of the value of and its D's at xo. The Rn is here yielded as a definite integral, from which form the other forms, as Lagrange's,

fluens, 1639, Napier fluxus, 1614. Clavius fluere, 1574); the velocity he writes and calls fluxio-our Derivative (as to the time t). Momentum varies as fluxion, is written xo, and corresponds to our Differential x (incrementa indefinite parva). This treatise seems to have been revised after 1673, hence