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 . hyperbola on which it is passes through(K,L) or (M,N). As yet we have only considered sections made by planes passingthrough the normal; we shall now suppose a section which declines fromthe normal by an angle v. As the theorem
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 . hyperbola on which it is passes through(K,L) or (M,N). As yet we have only considered sections made by planes passingthrough the normal; we shall now suppose a section which declines fromthe normal by an angle v. As the theorem we are now going to proveis isolated, I shall give a demonstration of it which assumes the infinitelysmall arcs of the sections to be parts of the circles of curvature, leavingthe student to try if he can express the equations of the sections, andthence determine the curvatures in the usual manner. Let OX be a line in the tangent plane, and take it as the axis of x:let OM be the normal section passing through that tangent, and let PObe an oblique section in the plane PNOA, making with ZOMN anangle AOZ = y. Let OQ be the projection of the section OP on theplane of XY. Then, since the equation of the surface is 2~=Rz2+2S xy + T^2+&c.; and since ON=#, we have 2NM = Rz2-f &c, (since y—0 for all points in- APPLICATION TO GEOMETRY OP THREE DIMENSIONS. 433fa. OM.) Again, since ON is tangent to OQ, NQ diminishes withoutlimit compared with ON; so that 2S xy and Ty2 are of the third and fourth order, or 2PQ=Rz2+ Consequently the limit of PQ: MN is unity, or PQMN approaches without limit to the form of a OP and OM for small arcs of circles, their diameters are thelimits of ON2: NP and ON2: NM, and diam. of OP : diam. of OM islimit of NM: NP, which as PMN approaches to a right angle, hascos PNM, or cos v for its limit. Hence, if OZ be the diameter of curva-ture of the normal section, and ZAO a circle with OZ for diameter, OAis the diameter of curvature of the oblique sec
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