CHAPTER X.

ON PROPERTIES OF PLANE CURVES, AS DEFINED BY EQUATIONS REFERRED TO RECTANGULAR COORDINATES.

SECTION 1.- Single tangents and normals, and their properties.

213.] In the following investigations it will be convenient to use sometimes the explicit, and at other times the implicit form of the equation to a curve in terms of two variables x and y. The general forms I shall take to be

y = f(x);

u = F(x, y) = c;

(1) (2)

and their differentials and derived functions will be expressed by the symbols which have been used in the former part of the volume; sometimes also the equations will be given in terms of three variables, as in Art. 208.

And as we are about to exhibit certain properties of geometrical space by means of Differential Calculus, I must say a few words on the mode by which a branch of the science of number is applied to the proof of geometrical truths. Differential Calculus does not of itself yield geometrical results: its results are numerical; but if a geometrical truth is so imagined as to be capable of expression in an analogous algebraical form; as, for instance, if we so imagine the fourth proposition of the second book of Euclid, as to express it in the analogous algebraica) form (a+b)2= a2+2ab+b2; or if a geometrical definition is so framed that we can express the numerical analogue of it with algebraical symbols; then we have certain algebraical formulæ, which are numerical truths, from which certain other numerical truths, contained explicitly or implicitly in them, may be deductively inferred. For this purpose of deductive inference, we are about to employ the Differential Calculus. It is evident therefore that the first step in the inquiry is, so to imagine the geometrical truths, or so to frame the geometrical definitions, that they may be translated into corresponding algebraical expressions; this has been done to a certain extent

in Section 3 of the preceding chapter. Thus they take an algebraical form, and become the subject matter of Infinitesimal Calculus and we are hereby enabled to deduce from them many properties, which are again to be translated into geometrical propositions. In accordance with this mode of inquiry many of the following Articles will begin with a geometrical definition.

It is true also that we may sometimes begin with an algebraical equation; in that case however the equation is only the symbolical expression of a geometrical proposition, which has been, or is capable of being, geometrically imagined. As, for instance, we may investigate certain geometrical properties of the curve whose equation is (31), Art. 201; but that equation is only the symbolical expression of the geometrical definition of the cycloid.

214.] We propose to find in the first place the general equation of a tangent to a given curve at a given point, which we define as follows:

The tangent to a curve at a given point is that straight line which passes through the point, and another point on the curve which is infinitesimally near to the former point.

Let & and be the current coordinates of the tangent line; and let (x, y) be the point of contact on the given curve; and let us suppose the straight line at first to pass through it, and through the point (x+▲x, y+ay), which is at a finite distance from it. Then we have the three equations

whence

A&+ BN + C
= 0

Ax+BY+ c = 0 ;

A.Ax+B.AY = 0

(§ —x) ▲y — (n—y) ax = 0.

(3)

(4)

Suppose the two points to approach infinitesimally near to each other, in which case ay and ar become respectively dy and dr, and the line whose equation is (4) becomes a tangent; and we have

(E—x) dy — (n—y) dx = 0,

Thus, for instance, if the equation to the parabola is

(7)

(8)

215.] If the equation to the curve is given in the implicit

and if this substitution is made in (5), the equation to the tangent becomes,

If the equation to the curve is a homogeneous function of n dimensions; then by the property of such functions, proved in Art. 82, equation (112), we have

and the equation (11) to the tangent becomes,

(12)

(13)

Generally, either (11) or (13) is the most convenient form for

the equation of the tangent.

Thus if the equation to the ellipse is given in the form

and as the equation is homogeneous and of two dimensions,

n = 2; and (13) becomes, after division by 2,

=

216.] Again, let the equation to the curve be given in terms of three variables, as (49), Art. 208, and be homogeneous, and of the form r(x, y, z) 0; let the coordinates to any point on a line be έ, n, C, and let this line pass through two points at an infinitesimal distance apart, viz. (x, y, z), (x + dx, y + dy, z + dz) ; then we shall have the following system of equations;

§ (y dz― z dy) + n (z dx − x dz) + 8 (x dy − y dx)

But by reason of the equation to the curve, and by reason of the homogeneity of that equation, we have

which is the equation to the tangent.

Thus, if the equation to a conic in the homogeneous form in terms of three variables is

Ax2+By2+cz2+2E y z +26 zx+2н xу = 0,

the equation to its tangent at the point (x, y, z) is

Ê (AX + GZ+HY) + n (By + Ez + H x) + Č (CZ+GX+EY) = 0, and which takes the ordinary form, if (= z = 1.

217.] To find the equation to the normal to a plane curve at a given point. The normal is defined as follows:

The normal to a plane curve at a given point is the straight line perpendicular to the tangent, and which passes through the point of contact.

Let έ, n be the current coordinates to the normal, and let

η

(x, y) be the point of contact of the tangent; then the equation to a line passing through (x, y) and perpendicular to that whose equation is (5), is

dx

n-y=

dy

which may also be written in the form

(n-y) dy + (x) dx = 0.

(19)

(20)

And if the equation to the curve is an implicit function, it

Hence also it follows, that the equation to a line passing through the origin, and perpendicular to the tangent, is

by means of which, in combination with equation (11), and that to the curve, the locus of the point of intersection of the tangent with the perpendicular on it from the origin may be determined.

Thus if the equation to the rectangular hyperbola is xy = k2, the equations to the tangent and to the perpendicular on the tangent from the origin are respectively

between which and the equation to the curve, if we eliminate x y, we have

and

¿2+n2 = 2k (§n) *.

The equation to the analogous curve with reference to the ellipse