A Treatise on the Differential Geometry of Curves and Surfaces

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Ginn, 1909 - Geometry, Differential - 474 pages
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Contents

I
1
II
52
III
70
IV
114
V
152
VI
189
VII
226
VIII
270
IX
321
X
351
XI
373
XII
392
XIII
426
XIV
446
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Page 219 - ... because the distances which we measure are too small, is often discussed. Gauss, in order to decide this question experimentally, measured as exactly as possible, and taking into account all possible sources of error, as large a triangle as possible (Brocken, Inselberg, Hoher Hagen), but could not find a difference between the sum of the angles of the triangle, and two right angles, which exceeded the limits of error. We shall return almost immediately to the question as to how far experience...
Page 241 - A surface which can be generated by the motion of a straight line is called a ruled surface. The infinitude of straight lines which thus lie on the surface are called its
Page 112 - In any triangle, the sum of the three angles is equal to two right angles, or 180.
Page 58 - Find the volume of the tetrahedron formed by the coordinate planes and the plane S + 5 + S-1' (7) F'S- 127- where a, b, c are all positive.
Page 403 - if a surface of reference of a normal congruence be deformed in such a way that the directions of the lines of the congruence with respect to the surface be unaltered, the congruence continues to be normal'.
Page 121 - D'dv) = 0 , (16) R 1 gndu+gndv Dndu + D'dv (17) D'du + D&dv = 0 The directions, for which R is extremura, are obtained from the equation (17). Then the two systems of curves passing through a point on the surface determine the directions at the point for which the radii of r-normal curvature have their maximum and minimum values. These curves are called the r-lines of curvature of the first kind and their tangents at a point the r '-principal directions of the first kind at the point.
Page 209 - ... be stated as follows : The excess over 180 of the sum of the angles of a triangle formed by shortest lines on a concavo-concave...
Page 332 - On the integration of a partial differential equation of the second order of the hyperbolic type, with more than two independent variables, by MR d'Adhemar.
Page 101 - If X, Y, Z are the coordinates of a point on the hodograph, we have for the relation between the two curves, 10,NY dx „ dy
Page 56 - AC2, and B3C3 all lie in one plane which is called the tangent plane to the surface at the point A. The point A is the point of contact. Since ADt and AD2 are any curves of the surface, it follows that, in general, the tangent at A to every curve of the surface through this point will lie in the tangent plane. Therefore, the tangent plane will contain all straight lines tangent to lines Fio.

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