An elementary treatise on coordinate geometry of three dimensions . 0. 282 COORDINATE GEOMETRY [CH. XIV. The equation may be further transformed. We have af—a(f) = Bz2-Cy2-((x.~a) = 0, etc. Hence the equation may be written ?£x3BC(a. - a) = ^BCiBz* - Cy2), = -(Bz2- Cy2)(Cx2- Az2){Ay2 -Bx2),= -()(P-b)(y-c),BCx*$ CAf-g ABz*£ , + ; ;+; + 1=0. ()S-6)(7-C) (y-cXa-ar («)(£-&)Ex. 3. Shew that at (x, y, z), a point of intersection of the three confocals,x2 y2 ?+- y a2+b2 + c2 l> a2 + \ b2 + X^c2 + X * a2 + fx 62 + /x c2 + /x +- = 1, ?/ +- 1, the osculating plane of the curve of intersection
An elementary treatise on coordinate geometry of three dimensions . 0. 282 COORDINATE GEOMETRY [CH. XIV. The equation may be further transformed. We have af—a(f) = Bz2-Cy2-((x.~a) = 0, etc. Hence the equation may be written ?£x3BC(a. - a) = ^BCiBz* - Cy2), = -(Bz2- Cy2)(Cx2- Az2){Ay2 -Bx2),= -()(P-b)(y-c),BCx*$ CAf-g ABz*£ , + ; ;+; + 1=0. ()S-6)(7-C) (y-cXa-ar («)(£-&)Ex. 3. Shew that at (x, y, z), a point of intersection of the three confocals,x2 y2 ?+- y a2+b2 + c2 l> a2 + \ b2 + X^c2 + X * a2 + fx 62 + /x c2 + /x +- = 1, ?/ +- 1, the osculating plane of the curve of intersection of the first twois given by X3/(a* + fl) | y/(62 + /x) , zz(c2 + [x)_1 -+- -+- aa(a2 + A) b2(b2 + k) c2(c2 + X) Ex. 4. Prove that the points of the curve of intersection of thesphere and conicoid rx2 + ry2 + rz2 = l, ax2 + by2 + cz2 = l,at which the osculating planes pass through the origin, lie on the cone t xi+ yi + t b-c c—a a—o = 0. 193. The principal normal and binormal. There is aninfinite number of normals to a curve at a given point, A, Ap. Fig. 54. on it, and their locus is the normal plane at A. Two of thenormals are of special importance, that which lies in theosculating plane at A and is called the principal normal, and §193] PRINCIPAL NORMAL AND BINORMAL that which is perpendicular bo the osculating plane and isCalled the binorraal. In fig. ~>4 AT is the tangent* AP theprincipal normal, ab the binormal; the plane atp is th<-osculating plane, and tlie plain- ABP is the normal plane ABT is called the rectifying plane. We shall choose as the positive direction of the tangentat A the direction in which the arc increases, and as thepositive direction of the principal normal, that towardswhich the concavity of the curve is turned. We shall thenchoose the positive direction of the binormal so that thepositive directions of the tangent, principal normal andbinormal can be brought by rotation into coincidence withthe positive directions of the x-,
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