An elementary treatise on coordinate geometry of three dimensions . c2, c3 Ex. 5. Find the curvature and torsion of the spherical indicatrixof the direction-cosines of the tangent are l2, m2, n2, (§ 197), and if So. is an infinitesimal arc, Lt o-r = l. by Hence, if the curvature is —,Po p7=2(§)2(§199) 1 1 — o 2 If the torsion is — cr02= , H°, as in Ex. 1. (dpoV Wol) §204] EXAMPLES ON CUBVATUBfi ANh TORSION Whence we easily findvhere r° piper-pv) pj£ and & (It,1, Ex. 6. Prove that the radii of curvature and torsion of the Bpb f) fi^ -4- cr^ indicatrix of the binomials are -i- - and
An elementary treatise on coordinate geometry of three dimensions . c2, c3 Ex. 5. Find the curvature and torsion of the spherical indicatrixof the direction-cosines of the tangent are l2, m2, n2, (§ 197), and if So. is an infinitesimal arc, Lt o-r = l. by Hence, if the curvature is —,Po p7=2(§)2(§199) 1 1 — o 2 If the torsion is — cr02= , H°, as in Ex. 1. (dpoV Wol) §204] EXAMPLES ON CUBVATUBfi ANh TORSION Whence we easily findvhere r° piper-pv) pj£ and & (It,1, Ex. 6. Prove that the radii of curvature and torsion of the Bpb f) fi^ -4- cr^ indicatrix of the binomials are -i- - and . ,- . Jpt + o-2 <r(<rp -<rp) Ex. 7. A curve is drawn on a right circular cone, a., so as to cut all the generators at the same angle B. Shewthat its projection on a plane at right angles to the axis is an equi-angular spiral, and find expressions for its curvature and Fig. 57. Take the vertex of the cone as origin and the axis as :-axis. Let C,fig. 57, be the projection of P, the point considered, on the axis, andCP and OP have measures r and R respectively. Then if CP makesan angle 6 with OX, r, 6 are the polar coordinates of the projection ofP on any plane at right angles to the axis. From fig. 58 we obtain Whence dr=dR sin oc = dz tan a, ^dR = ds cos B = r dd cot B. J — = cot 8 sin a. dO, (1) which is the differential equation to the projection and has as integral fcs where k = cot B sin a. and A is arbitrary. 296 COOKDINATE GEOMETEY [CH. XIV. dzAgain, from (1), j-= cos a. cos /3, and therefore the tangent to the curve makes a constant angle y with the 2-axis such that cos y = cos a. cos/?. d2zWe have therefore -^=0, cr=pta,ny. Since t^: ,2-°> i?-\d#) +VrfsV
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