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 . ome—«VC^2 + 772) = ^»2»£=bv, or |2-l-?j2=6*:a2, £=&tan_l(?? :£). So that the locus of thecentres of spherical curvature is another screw, generated by the samehelicoidal surface, but having a cylinder whose radius is 62: a. Thet
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 . ome—«VC^2 + 772) = ^»2»£=bv, or |2-l-?j2=6*:a2, £=&tan_l(?? :£). So that the locus of thecentres of spherical curvature is another screw, generated by the samehelicoidal surface, but having a cylinder whose radius is 62: a. Thetwo screws, however, are in opposite positions; for if in the first twoequations we make £=0, thereby obtaining the equations of the curvein which the polar surface cuts the plane of (xy), we find that £ and r)are the values of the coordinates of the involute of the circle whoseradius is 62: a, with their signs changed. The polar surface is then theosculating surface of this new screw: and if b=a, the osculating andpolar surfaces of the given screw are the same, the latter having onlymade a half revolution about the axis of z. For the coordinates of the centre of circular curvature, we findzylt—yzti-=. — ah* cos v — a3 cos v, yxtt — a/y/y=0, xztl — zxltz=. — a3 sin v—a62sinu, whence if X, Y, Z be the coordinates of this centre, wehave. X — a cos v— — a cos v cos v, Y—a sin u= —a sin v- a Z-bv=0; sin v, giving the equations of the same screw which is the locus of the centres ofspherical curvature. Looking now to the coordinates of the latter, wefind s=0, and -xllsi=—ab (a2+ bs) cos v, -ylls2=-ab(ai + bi)sinu, — zt/sai=01 giving for the values of Xu Yt, ZL the coordinates ofthe centre of spherical curvature, precisely the same as for the coordinatesof the centre of circular curvature. And the radius of spherical curva-ture is found to be a+, and the radius of circular curvature the APPLICATION TO GEOMETRY OF THREE DIMENSIONS. 417 same. The
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