+ 1 where a, ß, are subject to the condition a B2 a ? (2). 62 Differentiating with regard to the parameters, we have у B d8 = 0, a? 72 a da + If λ : a = 2, B = y; { and therefore, from (2), we have for the equation to the envelop a = X a, x? + y2 = 1, which is the equation to an ellipse similar in form to either of the original ones, its axes being of twice the magnitude. Again, if x = -1, we have from (3), a ( y = 0, which are the equations to a point, viz. the centre of the fixed ellipse, through which it is evident that all the moveable ellipses pass. X = a Intersection of Consecutive Normals to a Curve. 147. The equation to the normal at any point x, y, of a curve f (x', y') = 0, will be du du (a' - x) (y' - y) (1), dy dx where u = f(x, y). Differentiating (1), considering x as a vari du du able parameter, of which y, and are functional, we get dx' dy d2u d’u du dy? dy du dy - dy (2). d.c? dx dx + dy dx dx dy dx + dx dy From (1) and (2), and the differential of the equation ů = 0, we see that du? du? x y - Y dx * dy ? 2 dy dyž dx? dæ dy dx dy 'dx dy* the values of x', y', the coordinates of the intersection of two consecutive normals, coinciding with their values obtained otherwise in Art. (146). + CHAPTER IX. DIFFERENTIALS OF AREAS, VOLUMES, ARCS, AND SURFACES Differential of an Area. 148. Let AB be any portion of a curve referred to rectangular axes Ox, Oy. Let PM, QN, be the ordinates of two points P, Q, of the curve, very near to each other. Let AC be the the ordinates of A. Draw PR and QS, at right angles to QN and MP produced. Let OM = x, PM = y, ON = x + 8x, X , QN y + dy ; let A = the area ACMP, and A + DA area ACNQ. Then, since the area PQNM is evidently less than SQNM and greater than PRNM, it is plain that yox < 84 < (y + dy) 8x, DA y <y + dy. 8x Proceeding to the limit, when dx and dy become less than any assignable magnitudes, y + dy = y: hence ultimately, replacing small increments by differentials, we see that dᎪ dx or dA = ydx - Y, or < Differential of a Volume of Revolution. 149. Conceive a surface to be generated by the complete revolution of the curve AB, in the diagram of the preceding Article, about the axis of x. Let V represent the volume generated by the area ACMP, and V + 8V that generated by the area ACNQ. Now it is shewn in ordinary treatises on Trigonometry that the area of a circle is equal to 1 . (radius)', where a is the circular measure of 180°: hence the areas of the circles generated by the revolution of the ordinates PM, QN, will be equal to my?, 7 (y + dy), and therefore the volumes of the thin cylinders generated by the revolution of the areas PRNM, SQNM, will be equal to πgδα, π (y + δυ) δε. , (y + dy But it is plain that 8 V is greater than the former and less than the latter of these cylinders : hence πgδα < δY < π (y + δυ)? δε, SV or Now ultimately, when &x and dy become less than any assignable magnitudes, my’ and 7 (y + dy), the limits of the value of V become equal to each other: hence, replacing indefinitely dr small increments by differentials, we have dV try?, dx Differential of an Arc. 150. Let the chord of the arc PQ in the figure of Art. (148) be denoted by c; let arc AP s, arc PQ ds. Then 2 dy 1 + с = 1: Now ultimately, when 8x is diminished without limit, we know by the 7th Lemma of Newton's Principia that ds hence, replacing infinitesimal differences by differentials, we have ds? dy? 1 + dx2' ds” = dx? + dy’, a relation which has been already established in Art. (100). dx2 or Differential of a Surface of Revolution. 151. Let S, S+SS, denote the areas of the surfaces generated by the revolution of the arcs AP, AQ, of the curve AB, about the axis Ox; then SS will represent the area of the surface generated by the arc PQ. From P, Q, draw Pp, Qq, parallel to the axis of x and each equal to the arc PQ. Let AP = 8, PQ = ds. Then, if Pp, Qq, revolve about Ox = together with the rest of the diagram, they will generate two thin cylinders. The length of each of these cylinders will be equal to the curvilinear distance between the circular ends of the surface generated by PQ; but the average radii of the circular sections of this surface will evidently be greater than those of the former and less than those of the latter cylinder. Hence it is manifest that, 2tyds and 27 (y + dy) ds being the surfaces of the two cylinders, 2tyds < 8S < 27 (y + dy) ds, |