DERIVATIONS OF THE SCIENTIFIC TERMS IN GENERAL USE, TOGETHER WITH THE HISTORY AND DESCRIPTIONS OF THE OF HUMAN KNOWLEDGE. EDITED BY W. T. BRANDE, D.C.L. F.R.S.L. & F. LATE OF HER MAJESTY'S MINT; AND THE REV. GEORGE W. COX, M.A. AUTHOR OF A 'HISTORY OF GREECE,' 'THE MYTHOLOGY OF THE ARYAN NATIONS, ETC. NEW EDITION, REVISED. IN THREE VOLUMES. VOL. III. LONDON: LONGMANS, GREEN, AND CO. all Poles and Polars. The locus of the that point; indeed, generally, the polars of a harmonic centres of the (n-r)th order, taken point o situated on the primitive curve CD, with respect to a point or pole o, of the n inter- touch the latter in o, the common tangent being, sections of a given curve C, by a transversal of course, the polar line of o. The first polar which constantly passes through o, is called the Cn-1 of any point o is the locus of the points pth polar of o with respect to the given (primitive) of contact of all tangents drawn from o to the curve C. It is itself a curve of the (n-r)th primitive curve; so that n (n-1), or the numorder [HARMONIC CENTRES], and possesses very ber of intersections of C and C-1, is the important properties. Each point in the plane, maximum, and in general the actual number of therefore, has (n-1) distinct polars of the such tangents; this number, therefore, indicates orders n-1, n−2, .. 2, 1, respectively. the class of the curve. Should the primitive The last or (n-1)th polar is called also the curve possess multiple points, however, every polar line, and the last but one or (n-2)th the first polar will pass through them, and the polar conic, or quadric, the (n-3)th polar then (n-1) intersections of the latter with C. polar cubic, and so on. From the properties of will not all be the points of contact of harmonic centres it follows at once that any tangents from o. Thus if Cn has a double jolar of a point o is itself a polar of the same point at d, the 1st polar Cn-1 of o will pass point with respect to each of the systems of through it, and be there touched by the harpolars of higher order than itself. Thus the monic conjugate of do with respect to the two polar of o with respect to Cn is at one and tangents to Cn at d. This point d, therefore, the same time the (-1) polar of the 1st will count for two amongst the n (n-1) interpolar of C, the (r-2)th polar of the 2nd, the sections of Cn and Cn-1. If Cn have a cusp (-3) polar of the 3rd, and so on. As a at c, Ca-1 will not only pass through it, but special case the polar line of o with respect to will be there touched by the cuspidal tangent, C, is at the same time the polar line of o with so that c will count for three amongst the respect to all the other polars of Ca, con- intersections of Cn and C-1. We thus arrive sidered respectively as primitive curves. From at the result that the class of a curve of the the properties of harmonic centres, too, it fol-nth order, which has d double points and lows that the locus of the pole m whose rth k cusps, is n (n-1)-28-3к. This is the first polar passes through a fixed point o is the of Plücker's well-known equations. [SINGUin-r polar of o. Thus the polar line of o is LARITIES OF CURVES.] the locus of all points whose first polars pass through o, and further, the first polars of all points of a line constitute a pencil of curves of he (n-1)th order, passing through the same -1) points; these are the poles of that line. When the primitive curve is a conic, each Pant in the plane has, of course, but one polar, he polar line, and each line but one pole; in this case, too, the polar is simply the locus of he harmonic conjugate of the pole with repect to the points of intersection of the conic iy any transversal through this pole. The polar sa point on the conic itself is the tangent at VOL. III. 1 The subject of poles and polars will be found treated algebraically in Dr. Salmon's Higher Plane Curves, and geometrically by Steiner in Crelle's Journal, vol. xlvii. 1854, and by Cremona, in his Teoria geometrica delle Curve Piane (Bologna 1862). Poles and polars with respect to surfaces have a precisely similar definition. Their numerous and very important properties are treated with great ability by Dr. Salmon in his An. Geom. of Three Dimensions. Pole-axe. A long axe much used as an offensive weapon in the middle ages. It ap B |