that is, the asymptotes are two straight lines out of the plane of the paper, parallel to the axis of x, and at distances + 2a from it. An examination of which table shews that the curve is that drawn in fig. 35, where oc= CA= a; OB=OB′ = 2a. Ex. 8. Discuss the curve whose equation is Since the given equation is not changed when we write -x andy for x and y respectively, it appears that the curve is situated symmetrically in the four quadrants. Differentiating, we have therefore the equations to the asymptotes are y = + x; and, as the sign of the next term is positive, the curve lies above the asymptote in the first quadrant. When x = 0, y = 0; therefore the curve passes through the origin, at which point dy 0 = as appears from its second value given above, and is therefore to be evaluated. which implies that two branches of the curve touch the axis of a at the origin, both of which are out of the plane of the paper. The table of critical values is as follows: From 1 it appears that the curve passes through the origin, and has two branches, both of which are out of the plane of the paper, and which touch the axis of x; whence, as 2 and 3 shew, the curve recedes from the axis of x, until when x = ± a =OAOA', y = ∞, and there are two asymptotes parallel to the axis of y. For values of x beyond these lines, the curve is in the plane of reference, and returns towards the axis of x, until the ordinates reach minimum and maximum values when x = a√2, as is shewn by 4 and 5, whence it recedes again towards the asymptotes whose equations are y = ± x, and intersects them at infinity in a point of inflexion, as shewn by 6 and 7, the curve lying above the asymptote in the first quadrant, and being symmetrically situated in the others. Its course is traced in fig. 77, where OA = α, OB = √2a, BC = 2a, and where the dotted line represents the curve out of the plane of reference. If the equation to be discussed had been the branches of the curve which are in the plane of reference would have been out of it, and vice versa. The continuity of curve is remarkable in both cases. Ex. 9. Discuss the curve whose equation is whence it appears that the curve is symmetrical with respect to tion of the above quantities shews that the form of the curve is that drawn in fig. 78, the dotted branches being those out of the plane of the paper; OA = OB = BC = a, Ex. 10. Examine the Folium of Descartes, the equation to which is y3-3axy+x3 = 0. As shewn above in Ex. 3, Art. 235, the equation to the asymptote is y = Ꮖ a. Also at the origin there is a double point, as shewn in Ex. 3, Art. 249. = 0, if ay = x2; that is, if x = a (2), and y = a (4)*. dy dx dy dx Also =∞, if a ax= y2; that is, if x = a (4) 3, and y = a (2) *; and the curve does not extend beyond these limits; it is such as is delineated in fig. 63. In tracing curves of this kind involving circular functions, the arc, of which the trigonometrical function is given, is to be measured along the coordinate axis; in the present case along the axis of x, since sin x is involved in the equation, and the ordinates are to be constructed corresponding to the arcs or abscissæ thus measured; 7, we must remember, is the symbol for the arithmetical number 3.14159; and we must give to x such values as will render y a quantity capable of construction. Thus, in the equations above, let |