If through B and B' parallels N R and N' R' to A A' be drawn, and through A and A' parallels N'R and NR' to B B' be drawn, the diagonals R' R and N' N of this rectangle will be the positions to which the curve ultimately tends as it recedes from its centre.

For BA is to AC as any perpendicular drawn from a point in CR produced is to the distance of such perpendicular from C; and as this is the same ratio as the limiting ratio of PM to CM, it is evident that PM ultimately tends to equality with such perpendicular as CM is increased.

The line CN' produced has the same relation to the lower branch of the hyperbola.

The lines CY thus determined are called asymptotes. An asymptote in general is a tangent drawn to a point of the curve at an infinite distance, or, more strictly, it is the limit of the position of the tangent, the distance of the point of contact being supposed to be continually and indefinitely increased.

(688.) Hence it is apparent, that the curve approaches its asymptote continually, the distance between them decreasing without limit, but never vanishing.

(689.) An hyperbola may be described by a continuous motion in the following manner:- To the focus F' (fig. 243.) let a straight ruler F' L be attached by a

pivot, and to the other focus F let a flexible thread be attached by a pin, and let this thread be brought into contact with the ruler at P, and finally attached to its extremity at L. Let a pencil be looped in the thread at P, and held so that the thread shall be extended and the pencil pressed against the ruler. Let the ruler be thus turned slowly round the pivot F', and the pencil will describe an hyperbola whose transverse axis will be the difference between FP and F' P.

CHAP. XXI.

OF THE CURVATURE OF CURVES.

(690.) THE degree of curvature or flexure of a curve is estimated by the rapidity with which the point describing the curve departs from the tangent as it leaves the point of contact in either direction. A circle differs from all other curves whatever in having a perfectly uniform curvature throughout its whole circumference. If a tangent be drawn to any point in a circle, the arc of the circle, extending on either side of the point of contact, will be situated in exactly the same manner as an arc of the same circle would be with respect to a tangent at any other point. Thus if P (fig. 244.) be the point of

contact of a tangent PT, and P' be the point of contact of another tangent P' T', the arc PA on either side at P will be placed similarly to the arc P'A' on either side of P'. That this will necessarily be the case will be evident by considering that, if a segment A' A' be cut off by a chord, and the arc cut off be removed, and so placed that the point P' shall lie upon P, and the line P'T' on the line PT, the arc P'A' will lie upon the arc PA, and the same will be the case to whatever points in the circle the tangents may be drawn.

But if two circles have different magnitudes, they will then have different curvatures. Let PD and P D' (fig. 245.) be the diameters of two such circles, to which

PT shall be a common tangent at P. It is evident that the lesser circle will be contained within the greater, and that its circumference will depart from PT more rapidly than that of the greater circle. The curvature, therefore, of the lesser will be greater than the curvature of the greater.

(691) If the curvature be measured by the departure of arcs of equal length from the tangent, let m,m' be the extremities of two such arcs, and let mn and m'n' be the lines measuring their respective departures from the tangent. Drawm D and m' D'; also draw the chords m P'and m'P', which may be considered to coincide with the arcs, the latter being very small. It will be easy to show from the common properties of the circle, that the rectangle under D'P and m'n' will be equal to the square of the arc Pm', and the rectangle under DP and mn will be equal to the square of the arc Pm; but since

these arcs are equal, the rectangles under the diameters and the departures are equal; that is to say, in circles of different diameters the departures of equal arcs from their tangents are inversely as the diameters, therefore these diameters are inversely as the curvatures of the circles respectively.

(692.) The curvature of a circle being thus uniform, and the curvature of different circles being subject to unlimited variation by the increase or diminution of their diameters, the circle becomes the measure of the curvature of all other curves.

(693.) It will be evident, on inspection, that the curvature of an ellipse varies, gradually increasing from the extremities of its conjugate axis to the extremities of its transverse axis; the circle described with its conjugate axis as diameter, lies entirely within the ellipse, touching it at the points B Β ́ (fig. 246.); and since this circle