Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/63

INFINITESIMAL CALCULUS 53 Further considerations on the rectification of these curves will be found under the head of elliptic integrals. 169. Steiuer s theorem connecting the rectification of pedals and roulettes, analogous to that which connects their areas ( 165), may be here stated. It is as follows : If a curve roll mi a right line, the length of the roulette described by any point connected with the rolling curve is equal to the corre sponding arc of the pedal of the rolling curve, taken with respect to the describing point as origin. From this it is easily seen that the length of any arc of a cycloid is equal to that of a corresponding portion of a cardioid, and the length of a trochoid to an arc of a limaon. Again, if an ellipse be supposed to roll on a right line, the length of the roulette de scribed by either of its foci is equal to the length of the correspond ing arc of the auxiliary circle. Rectification of Curves of Double Curvature. 170. If the points in a curve be not in the same plane, the curve is said to be one of double curvature. Formula; for the rectification of curves of double curvature are easily obtained. Thus, if the curve be referred to a system of rect angular axes in space, we shall have rfs 2 = dx~ + dy~ + dz^ . Hence, if x be taken as the independent variable, we have f( , rfy 2 dz 2 *, f= /( 1 + ;/5 + T^> dx > J dx z dx*J and similar formula; when either y or z is taken as the independent variable. The equations of the curve are usually written in the form /(?;, 2/) = 0, 0(3, a) = ; that is, the curve is determined by the intersection of two cylinders. The values of l -^- and deduced from these equations dx dx have to be substituted in the foregoing integral. It is not difficult to determine a relation between the functions / and in order that the arc of the curve of intersection may admit of easy determination. The simplest class is where ( -^ ) = 2 ; for in this case dxj dx 1 + -r- )dx=x + z + const, dx/ For example, in the parabolic cylinder x* = 2py + const. , , dii x we have -^ . dx p ,1^ Accordingly, let and we get z = ^-= + const. , hence the length of the curve of intersection of the cylindrical surfaces is immediately determined. In general, when y=*f(x) is the equa tion of the first cylinder, and that of the second is represented by the equation z = J f (x) ] 2 dx + constant, the arc is determined by the above formula. 171. If we transform to polar coordinates by the relations x = r cos 6 sin 0, y = r sin 9 sin 0, z = r cos , we get ds z = dr 2 + r 2 d0 2 + r 2 sin 2 0d0 2 ; henee, for the rectification of a curve of double curvature we have dr 2 =/( 1 + r 2 -^- + r 1 sin 2 ; dr ; J V dr- dr*, 1 f( ^ d < = J( T ~ + d t The latter gives for the length of the arc of a curve on a sphere, of radius a, If be const. = a, the curve lies on a right cone ; and we have s=-/ | 1 +r- sin 2 a - n } dr. dr*J Cubature of Solids. 172. The method usually adopted, in seeking the volume of any solid, consists in supposing it divided by parallel planes into an in definite number of thin slices. Then in finding the volume we may in the limit consider each slice as an infinitely thin cylindrical plate ; and, consequently, represent its volume by the product of the area of the corresponding section into the indefinitely small distance between the parallel planes which bound it. Thus, if the points in the body be referred to a system of rectangular axes of coordinates, and the system of parallel planes be perpendicular to the axis of x, then, representing the area of the section at the distance x from the origin by A z , the volume of the solid will be represented by taken between proper limits. Adopting a similar notation, the volume of a solid may be repre sented by orfA t dz. In the case of a surface of revolution, the sections are drawn per pendicular to the axis of revolution. Thus, if any curve, situated in the plane xy, turn round the axis of x, a plane perpendicular t the axis cuts the surface in a circle. The area of this circle is iry* ; consequently the volume between two sections, corresponding to the abscissae a and b, is represented by ~, 4. (1) Suppose the ellipse + -i- = 1 to revolve round its axis of x t ct" u then the entire volume of the generated solid is r+a (2) If the parabola y = ax" revolve round the axis of x, the volume cut off by a plane at the distance X from the origin is o a X s + 1 XY 2 ,-x-"dx = ^n (3) To find the volume of the ellipsoid Hcre the section at the distance z from the origin is the ellipse ~o~ "t~ To" = 1 5 z 2 b 2 c 2 The area of this section A z is according!) 7 the volume of the ellipsoid is represented by r c i & lirab/ II -- j dz*= irabc. ~/Q C J (4) To find the volume of the surface generated by the revolution of a cycloid round its base. It is easily seen that the coordinates of any point on a cycloid, of radius a, are capable of being represented by x -= a(0 + sin 0), ?/=a(l + cos 0) . Hence the volume V generated is given by the equation v-2iray (i+c -M) - O (5) To find the volume of the portion of the paraboloid a; 2 y~ I m cut off by a plane drawn perpendicular to the axis of z. Here, the area of the section at the distance z from the origin is Hence, if c be the distance of the bounding plane, Consequently the volume is half that of the circumscribing cylinder. 173. Again, since any solid can be supposed divided into an in definite number of elementary parallelepipeds, the volume enclosed within any boundary may be represented by fffdxdydz , the limits being determined in each case by the nature of the pro blem.