sing: a + 2b 46(a+b) The radius of curvature of the epicycloid= the evolute is an epicycloid similar to the original curve, the radii of the base, and generating circle being respectively a? ab The equations between p and u, the centre of the base being the pole; u in the epicycloid, pre where e=a + 2b, - a 2 hypocycloid, p= (y e=a2b. The area contained by the axis, the radius vector, and the curve = (a +b)(a +26)(8 – sin o), in the epicycloid, hypocycloid. If a : b is a finite ratio, the curve may be represented by a finite algebraical equation. The equation of the cardioide, in which a=b, is y2 + (x − a)]* =a?{y® + (x − a)?}. If the diameter of the curve is the axis of X, and the cusp the origin, the polar equation is u=a(1 - cos y). The equation of the epitrochoid in which a=b, the origin being at the first point of contact, is The equation of the epicycloid with two cusps, in which b=ļa, is a*y* = (x2 + yo – aʼ). The equations of the hypotrochoid, in which b=ļa, are x=( a +b) cos , y=(la – 6,) sin 0, the equations of an ellipse, of which the semi-axes are ża + b, and la— bı. (Peacock, Ex. p. 192.) (43.) Spirals. The equation of the spiral of Archimedes, is u=a0; u the equation between р (a + u) a? . The equation of the Lituus is u*= 0 The first radius produced is an asymptote to the curve, and there is a point of contrary flexure. au' The équation between p and u; p= (a? + uo)? + (a® +u°)}-a. (ao +u*)* +a bu b c+ (co – tuo) or 0= log (a’ — U?) where c=(aż – 12). [2] p= [3] p= bu and a <b; then the polar equation is (a’ + u^)}' 0=log, (u* +co)} – C U b [5] The logarithmic spiral: p= -u, and a b log (a' – 6°) The angle contained between the radius vector and the 6 curve is constant, and = tan (a? — 6°)? The evolute is a similar spiral. (45.) The polar equation of the involute of the circle, is (u? -a)? = A + sec (46.) The tractory: The differential equation is The evolute of this curve is the catenary. . The equation of the syntractory is 6 + (° - 04 x=a log - (b^-^). y= If b<a, this curve has a point of inflexion, corresponding to b.at (2a +b)! If b>a, there is no point of inflexion, but when y=(ab), the tangent is perpendicular to the axis. If b<0, there is a point of inflexion, when -6. až y = (Peacock, Ex. p. 174.) (2a + b)}" (47.) The catenary. Let the curve be referred to its diameter, and a tangent at the vertex, and let s be the length of the arc; the differential equation is (c +8°) whence &+c=(s* + c), or s=(x* +2cx)}. =fc666 = loge с the equations between w and y are x+c=hceš +6=6), = = logo C с (Whewell, Mech. 111-3.) (48.) The elastic curve. Let the extremities of the elastic line be situated in the axis of y, at equal distances from the origin, and let a be the angle at which the curve passes through the origin; bø= -a' sin a, and c=a– bo, then the differential equations of the curve are a’ - 0 + 2012 a (c* — **)(2a’ –c? + x^)] Species [1]. Let , be very small: then, neglecting ca [2] : < 1. In this case the values of y and s can only be obtained by series. [3] ; = 1 In this case d y= and (a* a d.8= (a* — **)] Let l be the length of the curve, and h the distance of the extremity of the curve from the origin, then 121 19.32 1 l= 1+ V2 22.42 12 3 1 12.325 1 h= 1+ + + &c. V2 22 2 2.42 3 4 th=na”. πα To + + &c. &c.}; sc. πα. [7] ; =V2. The axis of y is an asymptote, and the equation of the curve is |