η, then, n,, in (59) are subject to the condition (60.) And as the pole moves along the directrix, so the position of the first polar continuously varies; and as the variation, corresponding to an infinitesimal variation of the pole, is infinitesimal, so the several and consecutive polars intersect, and in their intersections envelope another curve: this result is of course true for each system of polars, but at present we are considering only the first polars, and are inquiring into the envelope of these. If the directrix is a straight line, it has already, in Art. 227, been shewn that the envelope is (n-1)2 points. 319.] The degree of the first polar envelope may be determined as follows. Let us assume the degrees of the base-curve and of the directrix to be respectively n and r; then, as the position of the pole varies continuously, let us take the -, 7-, and -differentials of (59) and (60): so that we have and as no other relation is given between d§, dn, de, we have whence we have two equations in terms of x, y, z, έ, n, ; which are of n 1 dimensions in terms of x, y, z, and of r−1 dimensions in terms of έ, n, . If we eliminate έ, n, ¿ from these two equations and from equation (60), the resultant is of the order r (r−1) (n − 1) in terms of x, y and z; and therefore this is the degree of the first polar envelope. Thus if the directrix is a straight line, r = 1, and the degree of the polar envelope is zero: that is, as we have shewn in Art. 227, all the several first polars intersect in (n-1)2 points. If r = n = 2, that is, if the basecurve and the directrix are conics, the first polar envelope is also a conic. Let us consider one or two examples of the formation of these first polar envelopes. In the first place, I will take the base-curve to be the central conic whose equation is PRICE, VOL. I. 3Q } x2 y2 z2 + + = 0; a2 b2 C2 (64) and I will assume the directrix to be the straight line whose the pole being (§, n, 5). Now the first polar of (64) is (65) of which line the envelope is to be found, the pole (§, 7, 5) being subject to the condition (65). Therefore from (65) and (66) we have a2 A de + B dŋ + c d = 0, x= ▲ a2, c2 which are the equations to a point; and therefore all the first polars of (64) with reference to a pole situated on the straight line whose equation is (65) pass through the point whose coordinates are given in (67). Again, let us suppose the directrix to be a curve of the second degree, of which the equation is A §2 + B n2 + C52 + 2 En$+2 G 8§ + 2H & n = 0; (68) and let us suppose the base-curve to be the central conic (64), as in the preceding example; then the equation to the polar of (64) is yn 25 (69) of which the envelope is to be found, g, n, being subject to (68); whence we have (Aέ+Hn+Gg) d§ + (Hέ+Bn+EC) dn + (GE+En+cs) d$ = 0, and similar values are true for n and : substituting these in which is an equation of the second degree, and therefore represents a conic. Thus the envelope of all the first polars of a pole which is on a conic with respect to the central conic (64) is the conic (72); that is, as the pole moves along the conic (68) the envelope of the first polars with respect to (64) is another conic. 320.] The mutual relation of these two conics to each other by means of the central conic requires farther investigation. In the first place it is evident that to every point of the former conic a tangent of the latter corresponds: also let us take two points on the former conic infinitesimally near to each other; then the straight line passing through those two points is the tangent to the conic at the point: but the polars corresponding to all poles on this straight line pass, as we have shewn in the last Article, through the same point, that is, through the point of the second conic at which the two polars intersect which correspond to the poles on the former conic which are taken infinitesimally near to each other; and therefore this point is the pole of the tangent of the former conic. And a similar result is true of all points on this second conic; and therefore the former conic is the envelope of all the polars which correspond to the first polars of the points on the latter conic: or, in other words, the director-conic is the envelope of all the polars whose poles are on the conic envelope. The two conics therefore have reciprocal properties; each is the envelope of all the polars whose poles are on the other; and whatever peculiarities there are as to points on one conic, there are corresponding peculiarities as to the corresponding tangents of the other. Two conics thus related are called reciprocal polar conics. In the preceding investigation I have taken the base-curve to be a central conic. The process will be simplified, and the results will be scarcely less general, if a circle is the base-curve; and this is usually assumed to be the case unless it is expressly stated to the contrary. In this case, a = b = c. 321.] Now properties similar to these are capable of extension to other curves; let us suppose the directrix to be a curve of the nth degree; and let us suppose the first polars of all the poles, which are points on it, to be drawn, with respect to the base-curve; all these polars will envelope another curve, which is the polar envelope; and all points on this are also manifestly the poles of the tangents of the original director. Thus these two curves are reciprocal to each other; and are called reciprocal polars to a point on one a tangent on the other corresponds. And in these reciprocal relations very important properties have their origin all properties of curves become double: that which is true of certain points on a curve, is also true, mutatis mutandis, of the corresponding tangents of the reciprocal curve. Some few of these reciprocal properties are iudicated, although it is beyond the scope of my work to give a complete, or an approximately complete catalogue of them. : If the director curve is of the nth order, a straight line may cut it in n points; and as there is a different polar corresponding to each of these which touches the reciprocal curve, and as these polars will meet in a point which is the pole of the cutting line of the original curve, so to the reciprocal, from a given point, may n tangents be drawn: therefore the reciprocal polar is of the nth class; and the degree of a curve and the class of the reciprocal curve are the same. And since n(n-1) tangents may be drawn from a given point to a curve of the nth degree, so is (n-1) the degree of the polar reciprocal of a curve of the nth degree. If for one of two reciprocal polars a polygon is inscribed or circumscribed, then to cach vertex of the inscribed polygon the side of another polygon circumscribing the reciprocal corresponds; and to each side of an inscribed polygon the vertex of a circumscribed polygon corresponds. Thus, a vertex of an inscribed polygon is to be changed into the side of another polygon circumscribed about the reciprocal polar; and a side of an inscribed polygon into the vertex of a polygon circumscribed about the reciprocal polar. The Theorems of Pascal and Brianchon are thus reciprocally involved. It is plain also that the two reciprocal polars have the following corresponding and reciprocal properties. To a double point on one curve corresponds a double tangent on the other: to a multiple point of the kth order on one, a multiple tangent of the kth order on the other: to a cusp on one, a point of inflexion on the other. If many points on one curve are on a conic, the corresponding polars are tangents to another conic. As a curve of the nth degree is determined which passes n (n + 3) points, so is a curve of the nth class which through touches 2 n (n + 3) different lines. And as all curves of the nth degree which pass through Again, let n = the degree of a given curve, m = its class; d = the number of its double points; x = the number of its cusps; 7 = the number of its double tangents; i = the number of its points of inflexion; then, from (134), Art. 252, and (142), Art. 253, we have i=3n (n-2)-68-8k; and in the reciprocal polar these become n = m (m −1)-27-31, K=3m (m-2)-67-8i. (73) (74) (75) (76) This last equation however is not independent, but is involved in the other three. Thus, three independent equations are |