By Luther Pfahler Eisenhart

Created in particular for graduate scholars, this introductory treatise on differential geometry has been a hugely winning textbook for a few years. Its surprisingly precise and urban strategy encompasses a thorough clarification of the geometry of curves and surfaces, targeting difficulties that would be so much beneficial to scholars. 1909 variation.

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**Example text**

111) from the vertex, and Every point on the cone is at zero distance from the equations of the lines it is seen that the distance between call these generators of the any two points on a line is zero. We in space there are are their direction-cosines proportional to cone minimal straight an infinity of them ; Through any point lines. a. , where a is arbitrary. The locus of these lines is the cone whose vertex is at the point and whose generators pass through the circle at infinity. For, the equation in homogeneous coordinates of the sphere of unit radius and center at the origin is so that the equations of the circle at infinity are Hence the cone (111) passes through the circle at infinity.

INVOLUTES AND EYOLUTES 45 Hence the tangent to an involute is parallel to the principal norat the corresponding point, and consequently the mal of the curve tangents at these points are perpendicular to one another. As an example of the foregoing theory, cular helix, whose equations are x where a = a cos tt, = a sin u, z the involutes of the cir- = au cot 0, the radius of the cylinder and 6 the constant angle which the tangent to axis of the cylinder. Now is makes with the curve s = a cosec Q Hence the equations - a, u, /5, 7= - - -sin u,J cos w, cot & - cosec e of the involutes are csin0)sinic, From y we determine the last of these equations y = asinu it follows that the involutes are plane curves (au csin0)cosu, 2i = whose planes are normal to the axis of the cylinder, and from the expressions for x\ and yi it is seen that these curves are the involutes of the circular sections of the cylinder.

Arc *, (102) line touches the curve, as is If the coordinates 2, y, z of the coordinates of {- P are shown in M are expressed in terms of the given by CUKVES IN SPACE 42 differentiation with respect to where the accents denote s. When the equations of the curve have the general form the coordinates of P can where From is v be expressed thus = : t this it is seen that v is equal to the distance HP only when s the parameter. As given by equations (102) or (103), the coordinates of a point on the tangent surface are functions of two parameters.