Solutions of the problems and riders proposed in the Senate-house examination for 1857 - Tapa blanda
Sinopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1857 Excerpt: ...curve X sec-= s" a is constant, and that secg)' = approximately represents the evolute of this curve for the part near the origin. The length of the chord of curvature parallel to the axis of x is = 2a, a constant. Let f, rj, be the coordinates of the centre of curvature and c this chord. Then „ c c dx which therefore represents the evolute for the part near the origin. 6. Investigate the analytical conditions for the existence of multiple points in a curve of which the equation is u = 0, u being a rational function of x and y; and shew how the degree of multiplicity may be determined. Prove that, if du du _ dx2 + df1 at a double point, the coordinates of which are x, y, the two branches of the curve are at right angles to each other; and that, if the point be the origin, the equation of the tangents to the branches will be £, 77, being current coordinates of the tangent. At a double point the values of--are given by the equation / du d2u 0 dxdyj dy da? _ dx) 72vT + ' and therefore, if 0,, 02, be the directions of the two tangents du dx tan0,.tan0 „ = Tt" 1 2 d u which is the condition that the two branches may be at right angles. The equation of the two tangents through the origin is fa-tan0,.) fo-tan0,. £)-0, or rf-(tan 0, + tan 3) f17 + tan 0, tan 02f2 = 0 (I); but tan0, tan0 =-1, tan0, +tan=-: 1271 2 aw whence (1) becomes 8. Find the equation of the locus of tangent lines at a point (x, y, z) of a surface, the equation of which is u =/(%, y, z) = 0. Common tangent planes are drawn to the ellipsoids x2 y2 z2 „ x2 y z2 a b c 'a b c shew that the perpendiculars upon them from the origin lie in the surface of the cone (a-a') x + (b2-b'2) y + (c-c'2) a2 = 0. The equation of the tangent plane at the point (xyz) of the fir...
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Reseña del editor
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1857 Excerpt: ...curve X sec-= s" a is constant, and that secg)' = approximately represents the evolute of this curve for the part near the origin. The length of the chord of curvature parallel to the axis of x is = 2a, a constant. Let f, rj, be the coordinates of the centre of curvature and c this chord. Then „ c c dx which therefore represents the evolute for the part near the origin. 6. Investigate the analytical conditions for the existence of multiple points in a curve of which the equation is u = 0, u being a rational function of x and y; and shew how the degree of multiplicity may be determined. Prove that, if du du _ dx2 + df1 at a double point, the coordinates of which are x, y, the two branches of the curve are at right angles to each other; and that, if the point be the origin, the equation of the tangents to the branches will be £, 77, being current coordinates of the tangent. At a double point the values of--are given by the equation / du d2u 0 dxdyj dy da? _ dx) 72vT + ' and therefore, if 0,, 02, be the directions of the two tangents du dx tan0,.tan0 „ = Tt" 1 2 d u which is the condition that the two branches may be at right angles. The equation of the two tangents through the origin is fa-tan0,.) fo-tan0,. £)-0, or rf-(tan 0, + tan 3) f17 + tan 0, tan 02f2 = 0 (I); but tan0, tan0 =-1, tan0, +tan=-: 1271 2 aw whence (1) becomes 8. Find the equation of the locus of tangent lines at a point (x, y, z) of a surface, the equation of which is u =/(%, y, z) = 0. Common tangent planes are drawn to the ellipsoids x2 y2 z2 „ x2 y z2 a b c 'a b c shew that the perpendiculars upon them from the origin lie in the surface of the cone (a-a') x + (b2-b'2) y + (c-c'2) a2 = 0. The equation of the tangent plane at the point (xyz) of the fir...
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