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.

~ 122.

Geometrical discussion.

207

Z - a (z2 + - a)c- 2~b
2 (,zc-+ b)' 2 (zVc+ b)
But these are the formulas of the first substitution in ~ 120. Both
the other substitutions of that section are accounted for by analogous
considerations. By considering the pencil of rays at any point of the
conic, we obtain all possible substitutions by which the integral
is made rational. The study of algebraic integrals in general first
gains connexion and perspicuity by the geometry of algebraic curves;
this was first brought out in the fundamental works of A ronho d and
Clebsch (Journal f. Math., Vol. 61. 63. 64).
Regarding the present problem we have still to remark: the two
fundamental integrals to which every other was reduced, were:
dXv dx
fd andJ -, where y2 a + 2bx + cx2.
(Y J ^-Q)Y - -
Now considering one of the solutions we established for them, ex. gr.:
f -- ( + cx + gag) + c,
fq dx _f~1 l \Q) VR - A(Q) - '((Qe) (X - } +c,
we perceive that the integral function does not become infinite at the
points y = 0. These intersections with the axis of abscissa, at which
the tangents to the conic are parallel to the axis of ordinates, are
branching points of the function y but are not infinities for the
integral function. On the other hand, in the first formula the argument of the logarithm is infinite when x becomes infinitely great.
For the hyperbola there are two real points in which x = oo and
for the ellipse two complex points.
In the second formula when x == the argument of the logarithm
vanishes, and therefore as:
(kR_^f(e) --- f'() (- -)0 = (b + cx) ).f()) =
-x e )N)
Qthl Te oe tl itegl
the logarithm itself is iinfinite. The one fundamental integral is
logarithmically infinite at the two points at infinity upon the conic5
the other at its two points upon the right line x = p. This connexion between the two integrals becomes evident, when we can treat
the equation of the line at infinity like that of any other right line.
We can do so most simply by using homogeneous coordinates.
Putting x = x, - y the equation of the line at infinity
X3 X3
is x3 -= 0; that of the conic is x22 = ax2 +- 2bxlx3 + -ex1; and