218.] Let as be the distance between the two points on the curve through which the cutting line of Art. 214 passes, that is, let it be the length of the chord joining them; then and let the two points approach infinitesimally near to one another; in which case, according to the notation of Art. 17, ax, ay, as become respectively dx, dy, ds, and we have ds2 = dx2 + dy2; (23) and ds becomes the distance between these two points, which are infinitesimally near to each other; that is, it becomes an element of the curve, or an infinitesimal arc; or, as we shall call it, a length-element of the curve: it is in fact the small portion of the tangent line which is common to the tangent and the curve, the tangent indeed being the length-element produced. Or, under another mode of considering the curve, that is, of conceiving it to be generated by a point moving according to a given law, ds is the distance between two successive positions of the point; and if these two positions are taken so near to each other, that only an infinitesimal instant of time has elapsed during the passage from one to the other, it is impossible to conceive but that the moving point has passed in a straight line from one to the other; the length of which straight line is ds. If then we use the character to symbolize the angle made by the tangent with the axis of x, that is, the angle PTM in fig. 47, we have, from equation (5), the last equality following from Preliminary Theorem I; the numerator and denominator of the preceding equalities having been squared and added, and subsequently the square root having been extracted. The preceding equations frequently render it convenient to deduce from the equation to a curve the relations between ds, dx, and dy: of which some examples, giving rise to differential expressions, are subjoined. Ex. 2. To find the relations between dx, dy and ds, in the equation to the cycloid. (a) Let the starting point be the origin; therefore, by equation (29), Art. 201, x = a versin-1 - {2ay—y°}‡ ; PRICE, VOL. I. У X X (B) Let the highest point be the origin; therefore, by equa Ex. 3. To find the relation between da, dy and ds, in the equation to the catenary. 219.] Let us now consider the general geometrical results which are contained in the preceding equations to the tangent and the normal. Let us assume the curve drawn in fig. 47 to be the typical form of all curves; of which PT is the tangent line at the point P, PG the normal line; MT the subtangent; MG the subnormal; or the perpendicular from the origin on the tangent line; or, OT respectively the intercepts of the axes of x and y by the tangent line; and the lines, PT and PG, the parts of the tangent and normal lines intercepted between the point of contact and the axis of x, are called respectively the tangent and the normal. Then, by the equations to the tangent, (5) and (11), the numerators of these last values generally admit of simplification by Euler's Theorem; but of this hereafter. Without however deducing the values of the other geometrical lines of the figure from the preceding equations, we will express them in the following manner; which is preferable, because it addresses itself more directly to geometrical construction and to the eye. Also by means of equations (27) and (28), (44) and therefore, if r (x, y) is a homogeneous function of n dimensions, 2 dr 2 dy Hereby also we may put in other forms the equation to the tangent given in equation (5), 220.] From the equations to the tangent and normal it appears, that whenever a or y is affected with+, such signs will remain in the equations, and therefore 7 and έ will be similarly affected; and therefore whenever the curve is out of the plane of reference, the tangent and normal also are. η Also, as is the trigonometrical tangent of the angle made dy with the axis of a by the tangent to the curve, at all points at dy which has a finite value the curve is inclined to the axis of dx dy dx x at a finite angle and if is positive, then x and y are simultaneously increasing or decreasing, see Art. 110, and the curve is such as the logarithmic curve of fig. 32, or the cycloid dy dx of fig. 40; and if is negative, as increases y decreases, and vice versa, and the curve is such as the equitangential curve in fig. 38. dy If = dx O, the tangent, and therefore the curve at the point dy of contact, is parallel to the axis of x; and if changes its dx |