The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . : (V+y//8+V) or dv J^+yt+z*). (*„x»+yl/y»+z/lz) : (xj+yj+zj) J the first two factors of which being —ds, we have (^//2+y//2+~//2) :(#,, #+:?/„ t//; 4-~//£) for what we may call the radius of flexure. We have not yet found an evo
The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . : (V+y//8+V) or dv J^+yt+z*). (*„x»+yl/y»+z/lz) : (xj+yj+zj) J the first two factors of which being —ds, we have (^//2+y//2+~//2) :(#,, #+:?/„ t//; 4-~//£) for what we may call the radius of flexure. We have not yet found an evolute of the curve, or a second curvewbose tangents are normals of the first. The two loci of circular andspherical curvature are not of this character. If any evolutes exist they 414 DIFFERENTIAL AND INTEGRAL CALCULUS. must lie on the polar surface, and not elsewhere, for all normals lie innormal planes, whence the intersection of two consecutive normals mustlie in the intersection of two consecutive normal planes, or on a polarline; that is, on the polar surface. And we can .obviously make aninfinite number of evolutes on the polar surface: thus, let P, Q, R, Sbe consecutive points of the curve, infinitely near, through which draw. normal planes giving 14V part of the polar surface: join P with anypoint 1 of its polar line, draw Ql and produce it to meet the succeedingpolar line in 2, and so on. We have then as many small arcs of anevolute, 1, 2, 3,4, as we can take points in the first polar line to joinwith P. Or, through every point of the polar surface one evolute passes,and only one. The question of finding an evolute is, therefore, reducedto that of drawing a curve on the polar surface, whose tangent shall alwayspass through the given curve. But since every tangent plane of thepolar surface cuts the curve somewhere, one condition is satisfied by themere circumstance of the curve lying on the polar surface, which makesits tangent lie in a plane cutting the curve.
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