182. Polar Equation of the Tangent. Let the polar co-ordinates of the point of contact be du a; and let U' be the value of for the curve at (1 a); that point. the form do The equation of any straight line may be written in A cos (0− a)+B sin (0− a), ..............(1) A and B being the arbitrary constants. Let this straight line represent the required tangent. du do By differentiation du do = — A sin (0− a)+B cos (0-a). ....(2) Now, since the tangent touches the curve, the value of at the point of contact is the same for the curve and α for the tangent. Hence, putting 0=a in equations (1) and (2), we have U=A and U'=B, whence the required equation will be u = U cos (0− a) + U’sin (0 — a). 183. Polar Equation of the Normal. .(3) The equation of any straight line at right angles to the tangent given by equation (3) of the preceding article may be written in the form Cu = U'cos (0 - a) — U sin (0-α), C being an arbitrary constant. This equation is to be satisfied by u=U, 0=a for the point of contact; therefore substituting we have CU = U', whence the required equation of the normal is. U' U u = U'cos (0 — a) — U sin (0 — a). 184. Class of a Curve of the nth degree. DEF. The number of tangents which can be drawn from a given point to a rational algebraic curve is called its class. Let the equation of the curve be f(x, y)=0. The equation of the tangent at the point (x, y) is af x + y f Эх +2 0, where z is to be put equal to unity after the differentiation is performed. If this pass through the point h, k we This is an equation of the (n-1)th degree in x and y and represents a curve of the (n-1)th degree passing through the points of contact of the tangents drawn from the point (h, k) to the curve f(x, y)=0. These two curves have n(n-1) points of intersection, and therefore there are n(n-1) points of contact corresponding to n(n-1) tangents, real or imaginary, which can be drawn from a given point to a curve of the nth degree.* It appears then that if the degree of a curve be n, its * Poncelet, "Annales de Gergonne,” vol. VIII. class is n(n-1); for example, the classes of a conic, a cubic, a quartic are the second, sixth, twelfth respectively. 185. Number of Normals which can be drawn to a Curve to pass through a given point. Let h, k be the point through which the normals are to pass. The equation of the normal to the curve f(x, y)=0 at X-x Y-y the point (x, y) is This equation is of the nth degree in x and y and represents a curve which goes through the feet of all normals which can be drawn from the point h, k to the curve. Combining this with f(x, y) =0, which is also of the nth degree, it appears that there are n2 points of intersection, and that therefore there can be n2 normals, real or imaginary, drawn to a given curve to pass through a given point. For example, if the curve be an ellipse, n=2, and the number of normals is 4. Let x2 +32 a2 b2 =1 be the equation of the curve, then is the curve which, with the ellipse, determines the feet of the normals drawn from the point (h, k). This is a rectangular hyperbola which passes through the origin and through the point (h, k). (2) g (2) on which lie the points of contact of tangents and the feet of the normals respectively, which can be drawn to the curve f(x, y)=0 so as to pass through the point (h, k), are the same for the curve f(x, y) = a. And, as equations (1) and (2) do not depend on ɑ, they represent the loci of the points of contact and of the feet of the normals respectively for all values of a, that is, for all members of the family of curves obtained by varying a in f(x, y) = a in any arbitrary manner. is called the "First Polar Curve" of the point h, k with regard to the curve f(x, y)=0; z being a linear unit introduced as explained previously to make f(x, y) homogeneous in x, y, z, and put equal to unity after the differentiation is performed. As this is a curve of the (n-1)th degree it is clear that the first polar of a point with regard to a conic is a straight line, the first polar with regard to a cubic is a conic, and so on. The first polar of the origin is given by If the curve be put in the form Un + Un-1+Un-2 + ... + u2 + u12+u1 =0 the first polar of the origin is Un-1+2Un-2+3Un-3 + ... + (n − 1)u2+nu ̧=0. the polar line of the origin has for its equation 188. The p,r or Pedal Equation of a Curve. In many curves the relation between the perpendicular on the tangent and the radius vector of the point of contact from some given point is very simple, and when known it frequently forms a very useful equation to the curve; especially indeed in investigating certain Statical and Dynamical properties. 189. Pedal Equation deduced from Cartesian. Suppose the curve to be given by its Cartesian Equation and the origin to be taken at the point with regard to which it is required to find the Pedal Equation of the curve. Let x, y be the co-ordinates of any point on the curve; then, if F(x, y)=0 be the equation of the curve, the equation of the tangent is хран Эх ду ƏF + z where z is as usual to be put equal unity after the differentiation is performed. If p be the perpendicular from the origin on the tangent at (x, y) we have |