An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

206

The integral of explicitly irrational functions. Bk. III. ch. III,

Ill. d 9_ dz __ 2i dz
J x -- piCPj ^
V -- 2
2 tan-'1 z +C,
x 2, -./
or I_ - = V —V_ tan + C.
This equation can also be generalised according to the above process:
we have
i f ___dx ______ _s _ tn, ______
III'. S( -QRP — c~ yi,(: A'tan- -- Q + C,
' - s)1/-1- -/( - y)(P-o) C P-Q (/-a).- -c
where a and 3 are the roots of the quadratic equation == 0.
Note: The integral jF(x, j/-) cdx can be discussed by the help
of geometrical considerations. Denoting the value:
VI/ = /a + 2bx +- cx by y,
the equation: y2 ==_ a + 2bx +- cx, referred to Cartesian coordinates
represents a curve of the second order; this is a hyperbola if c > 0,
a parabola if c = 0, an ellipse if c < 0, but, when at the same time
a c - b > 0 the curve is completely imaginary. The axis of abscissa
is an axis of the curve, to each value of x correspond two equal and
opposite values of y, or ~- V/.
The integral: fF(X, y)cdx is uniquely related to the curve, i. e.
to each of its points, real or complex, belongs one value of the
function to be integrated, for, the sign of the root is determined by
the point of the curve. Now our investigations have shown, that
such an integral extended along a conic can be transformed into a
rational one, that therefore the coordinates x, y, of points on the conic
can be expressed as rational functions of a variable z. If conversely
we assume this theorem, that is found in the projective geometry of
conics to be a fundamental theorem, the methods of treating the irrational
integral are simple deductions and lose all appearance of an artificial
substitution. Thus ex. gr. for the hyperbola y2 == a + 2bx- + cx2, c > 0,
the directions of the asymptotes are given by y2 - ex2 = 0. If now
we construct a system of right lines parallel to one asymptote, we
have for the equation of this system y +- xl/c= z, where z stands
for a variable. Each of these right lines meets the hyperbola only
in one finite point, its other intersection being always the same point
at infinity, and the coordinates of this single finite intersection are
expressed as the following functions of:'