А CO 2 and we shall find C=-4 pl p. Consequently AM = y. р V(99+ * mp)+1 pl (9+ Vyy +4 pp). 2 р 113. It may be remarked, if from A as a centre, and with the semi-transverse axis BA=} p, we describe an equilateral hyperbola BN, the space ABNQ will be sy v (yy + pp). Therefore AM p = ABNQ; from which Pe it follows that the rectification of the parabola depends upon the quadrature of the hyperbola, and reciprocally. Ex. III. In the ellipse, if we suppose the semi-transverse axis =1, we shall have y=bv (1-xx); and making ✓(1-6b), or half the dis N' tance between the foci =c, we obtain BM=8V(1—ccxx) an integral which ✓(1-xx) A А cannot be obtained by the preceding rules. We must therefore reduce the B expression into a series ; but for the pure pose of simplification, we shall only re N' D duce vi--ccxx); we shall then find BMES 1.3 1.3.5 2.4.6.8 n D M B В A P -C8 28-&c.} xli (1-xx By (61) it appears that s ✓(1-2X) 2i -r!!! (21-1) (-3)...3 x (1-XX 2i (2,-2)...2 And there re VII-IX) c? 304 3.5 C BM =(! 3.52.7 c8 &c.)S ✓(1-**) But the arc DNES consequently we know all the ✓(1-x)' quantities that enter into this series, of which the law is easily perceived. Let x=1, and we shall have co 8c4 А 34.5 c6 --&c.) AND. Therefore the pe2* 2.4 2.4.6* riphery of the ellipse is to that of the circumscribed circle as c 1.12 8 1.1.3 5 c -&c. : 1 (callingthe semi22 a 2.4" 2.4.63 am transverse axis a). This series will be very convergent when the 1 . 10 ference of the ellipse will be to that of the circumscribed circle as 0.997495 292 861 261 : 1 The rectification of the 'hyperbola may be found in nearly the rame manner. 8 a Ex. IV. The equation of the second cubical parabola is yo=as'. Consequently S V(#*+ý) = süv(1 + (1++c Making y=0, we have CE a, and any arc of this curve com. 27 8 puted from the vertex {(1+2—1} 27 Ex. V. In the cycloid j=ivi). Therefore Vix*+ý°)=id on; and taking the fluent, we obtain AM=2N ax=2 AN by (37) Ex. VI. In the logarithmic eurve yż=ay, and wiö?+92) = ✓ (yy +aa). Let v (yy taa)=x, and we shall have yy=z2-8, and again v (izi + y) = x+. -, of which the 4 and y aaż aa integral is z+ ,ord (aa+yy)—al {a+v(aa+ )} +C; zta y the expression for any arc of the logarithmic curve, in which C is easily determined. Ex. VII. In the spiral of Archimedes, if we make B AGFBN = x, CM = y, we we shall have Mr = ух N M P (yy + 2a* CN, whose parameter by making CQ=CM, and drawing the ordinate QN, we shall have CN=SOPIS ve+yy). Therefore CN'=COM. From which we may conclude, that there exists some analogy between the Archimedean spiral and the parabola. Ex. VIII. In the hyperbolic spiral, the arc Com=sy_v (66+yy). Therefore if we describe a logarithmic curve NK, whose subtangent == to that of the spiral, we shall have MOC= the infinite arc NK, by taking the ordinate NR=CQ= CM. But if we desire an expression for any arc of the spiral, or of the logarithmic curve contained between two ordinates y, y', we shall find it to be (6b+yy)-> (66+44)+619(6+V66+yy (b+V66+vv с DM=?=MT C+ (m+1) Hi or simply Example IX. In the logarithmic spiral cos Mmr : mr :: 1 : Mm that is cry::1: y B в Therefore ADM=; ON THE CUBATURE OF SOLIDS. 114. In order to find the contents of any proposed solid, we must conceive it to be decomposed into an infinite number of thin seg. ments or slices parallel to one another. Calling t the surface of one of these segments, wc its thickness or an indefinitely small portion of a line perpendicular to that segment; S () +C will be the content of the proposed solid ; there will then only remain for us to find the value of t in terms of x. 115. For example, let B be the base of the solid ; H its altitude, or the distance of the base from its vertex. If we suppose that the surfaces of these segments are in proportion to any power m of their B.x** distance from the vertex, we shall have HTM : B :: x": Conse Bux quently the solidity of any portion of the solid will be C+ m+1 m+1 B.x if the portion begins from (m+1) HTM' BH the vertex. Consequently the entire solid (because x coin B.X " m+1' cides with H). Therefore in pyramids this solidity =}BH, since m=2. 116. If any curve AM revolves about its axis AP, it will generate M a solid of revolution of which each section perpendicular to the axis N will be a circle, having for its expression myy, calling PM = y, and 7=3.14159, &c. Consequent BE ly the content of any solid of revolution =C+Sayye. Example I. In the sphere yy=2ac. Therefore the solidity of any spherical segment =mxr (a-4x), and the sphere =j 2a = two-thirds of the circumscribing cylinder. 66 Ex. II. In the ellipse yy= (2ax-xx). Consequently the solid aa m generated by its revolution about its transverse axis is to the circumscribing sphere as bb : aa; or it is = to two-thirds the circumscribing cylinder. 117. The solid which we have just considered is called an oblong spheroid ; and that which is formed by the revolution of an ellipse about its less or conjugate axis, is termed an oblate spheroid. It is easy to shew that this latter solid is also equal to two-thirds of its circumscribing cylinder. Consequently the oblong is to the oblate spheroid :: abb: aab:: 6:a. Ex. IIl. If a parabola of any degree whose equation is y=x** * revolves about its axis, it will generate a solid, having for its expression Saya = Txy? ; or which will be to the circum m + 2n scribing cylinder :: m : m +2n. Thus the common paraboloid, in which m=2, n=1, is the half of its circumscribed cylinder. Example IV. Similarly, if the hyperbola whose equation is y" x" mtn za revolves about its asymptote CP; by taking CD=AD-a, the solid described by the trapezium ADMP, will be expressed by (a}--xy?), and consequent- c 2n-m P ly the solid described by the indefinitely long space OADX, is to the cylinder described by ACDE :: m : 21—m; and in the common hyperbola it is equal to such cylinder. M m |