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 dφ 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” ρ is the radius of this circle. We have 1/ρ = |dφ/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= | { 1 + ( dy)2 }−3/2 d2y|,
ρ dx dx2
ρ = | r dr| = | p + d2p|.
dp dφ2

(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 ƒ(x) = 0, the function ƒ(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 ƒ(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 + ... + un = 0, where u0 is a constant, and ur 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 q1 x2 + ...,   or p2x + ½ q2 x2 + ...,

where q1, ... and q2, ... are determined without ambiguity. If p1 and p2 are real the two branches have radii of curvature ρ1, ρ2 determined by the formulae