Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/62

52 INFINITESIMAL CALCULUS If, now, x, y be the coordinates of with respect to a pair of rectan gular axes drawn through 0, we shall have p p - x cos co - y sin co ; therefore flit /-27I- ^TT A - A = i/ (x cos co + y sin a>) 3 <fco - x I p cos <ad<a-y / p sin a* rfw . yo yo yo But /2rr flit m flit _ 008*0 dca = TT , / sin 2 co du = ir , / sin co cos co dco = ; yo yo consequently A - A = ^-(x 2 + y" 2 ) -gx-hy , where p cos co dca , h=. Hence we infer that, if be fixed, the locus of , when the corre sponding pedal area A is constant, is a circle. All the circles obtained by varying the pedal area are concentric. Also the common centre is the point for which the pedal area is a minimum, and the pedal area with respect to any origin exceeds the minimum pedal area by half the area of the circle whose radius is the distance between the pedal origins. Many interesting results may be deduced from this theorem. When the curve is not closed, it is easy to prove, as was shown by Prof. Raabe (Crelle, vol. 1.), that the locus of the origin for pedals of equal areas is an ellipse. The corresponding theorems for the volumes of the pedals of surfaces were investigated by Dr Hirst ( Transactions of the Royal Society, 1863). In addition to other important generalizations, Dr Hirst has here proved, when the surface is closed, that the locus of the origin for equal pedal volumes is a surface of the second degree. Another remarkable theorem of Steiner s, on the connexion be tween the areas of pedals and of roulettes, may be stated here. When a closed curve rolls on a right line, the area between the right line and the roulette generated by any point invariably con nected with the rolling curve, in a complete revolution, is double the area of the pedal of the rolling curve, taken with respect to the gene rating point as origin. Hence it follows that there is one point in a closed curve for which the entire area of the roulette, described in A complete revolution, is a minimum. Also, the area of the roulette described by any other point exceeds that of the minimum roulette by the area of the circle whose radius is the distance between the points. Rectification of Curves. 166. The rectification of curves is based on the principle that the length of an arc of any curve is the limit to which the perimeter of an inscribed polygon approaches when each of its sides is conceived to diminish indefinitely. Hence, if the curve be referred to rectangular axes of coordinates, and if ds denote the element of the arc of the curve at the point (x, y), we shall have ds 1 and accordingly taken between the limiting points, i.e. , the extremities of the arc. In like manner if the curve be referred to polar coordinates we shall have We shall illustrate these formula; by a few simple cases. Ex. 1. In the ordinary parabola x* = 2py we have i/ ^_ . dx p Ex. 2. In the more general parabolic curve represented by x n *=py we have dx p This expression is capable of integration in a finite algebraical form (8 123) for the following values of In- 2, i. e. , when n is n r, 7 n t , fr 167. In illustration of the method of rectification in polar coor dinates, we commence with the spiral of Archimedes, r = a& . 1 / H ere s = - / (r* + a )$dr . This shows that the length of any arc of this spiral is equal to that of a corresponding arc of a parabola. This relation between the spiral of Archimedes and the parabola was discovered, according to Sir John Leslie, by Gregoire St Vin cent, before the middle of the 17th century (see Leslie s Geometri cal Analysis, p. 424). That a corresponding relation connected the parabola y n =px and the spiral r"" 1 = - pO was established by n John Bernoulli (Ada Erud., 1691). These results were extended by Lardner (Algebraic Geometry, p. 355), and in their general form may be stated thus : If from the equation to any curve in rectangular coordinates another curve in polar coordinates be formed, by making dy = dr and dx = rd#, then the len/jth of any arc of the second curve will be equal to that of the corresponding arc of the first curve. Also the sec- torial area of the second curve will be half the area bounded by the corresponding y ordinatcs in the first curve. These relations can be immediately established. As an example, the right line y = mx gives by this transforma tion the logarithmic spiral r = c mS . Hence we can always obtain a portion of a right line equal in length to any arc of this spiral, a result which is obvious otherwise. Again, from the ellipse ydy Hence the differential equation of the transformed curve is a dr de= --T- - , b ^-r 2 from which we get r= 6 cos t where 6 is measured from the line which corresponds to the major axis of the ellipse. Accordingly, the rectification and quadrature of this latter curve is the same as for the __^r ellipse. This can also be shown imme diately otherwise. 168. Whenever the pedal equation of a curve ( 165) can be found, there is an other general formula for its rectifica tion, which may be proved thus. In fig. 9 let ON be the perpendicular let fall on the tangent at any point P on a curve, and ON the perpendicular on the tangent at a consecutive point Q; and suppose ON =p , angle AON = co, arid PN = t . Then PQ-As, angle SON = Aco , A< = QN -PN. Hence d * lim PT + TQ <** - ij ln QP -FN = lim. . -= 11111. . ft co Aco (/a) Aco But PT + TQ + PN - QN = TN - TN ; ds dt TN-TN SN hence - =i im . =lnn. = ON = w. uco t ca Aco Aco Accordingly, if u l and co be the values of co corresponding to the extremities of the arc s, and t l} t the corresponding values of t, we have /-to)! S = t 1 -t + / pdca. y<oo This theorem is due to Legcndre. In its application it is well te observe, that 4?_lim.^ -Hm.TN -. (tea Aco For example, in the parabola we have dp a sin co COS 2 CO Hence, if s be measured from the vertex of the parabola, we have a sin co /" w dca = - T, -- ha/ cos 2 co JQ cos co . log tan ,- + -pr w -pr I 2 / Similarly in the ellipse,^ + ^ = 1 , we have^>= Ja" cos - co + 6 2 sin - co. Accordingly, the rectification of the ellipse depends on the integral J V 2 cc "COS-co + U sm- ca dca . Likewise the rectification of the hyperbola depends on the integral c^co.