General Investigations of Curved Surfaces of 1827 and 1825 |
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arbitrary auxiliary sphere circle coordinates correct to terms correspond cos² curved line curved surface denote Derivation of formula differential dx² equal equation expression Ə R₂ ƏR₁ ƏR₂ figure formula functions ƒ'p³ ƒ°² p² Gauss infinitely small integral curvature linear element mean curvature measure of curvature multiplying normal obtain p² q² p² q³ p³ q p³ q² p³ q³ plane preceding article q aq q ə q q¹² q² ² q² q quantities R₁ radius of curvature represents the direction right angles shortest lines side spherical straight lines suppose ť² theorem triangle values ар аф ах ду дв др дя эф
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Page 97 - The difference between any two consecutive convergents is a fraction whose numerator is unity, and whose denominator is the product of the denominators of the convergents.
Page 15 - The measure of curvature at any point whatever of the surface is equal to a fraction whose numerator is unity, and whose denominator is the product of the two extreme radii of curvature of the sections by normal planes.
Page 49 - ... conviction that, for all measurable triangles on the surface of the earth, they are to be regarded as quite insensible. So it is, for example, in the case of the greatest triangle of the triangulation carried out by the author. The greatest side of this triangle is almost fifteen geographical* miles, and the excess of the sum of its three angles over two right angles amounts almost to fifteen seconds. The three reductions of the angles of the plane triangle are 4".95113, 4".
Page 114 - ... to be convex, ie such that no side produced will enter the polygon. 157. Polygons that are mutually equiangular and equilateral are congruent, for they can be made to coincide. Ex. 379. Draw two mutually equiangular quadrilaterals that are not mutually equilateral. PROPOSITION XLI. THEOREM 158. The sum of all the angles of a polygon of n sides is equal to (n — 2) straight angles.
Page 52 - Germain defined as a measure of curvature at a point of a surface the sum of the reciprocals of the principal radii of curvature at that point, or double the so-called mean curvature.
Page 30 - The excess over 180° of the sum of the angles of a triangle formed by shortest lines on a concavo-concave...
Page 47 - F, a are again functions of p and q. The new expression for the measure of curvature mentioned above contains merely these magnitudes and their partial differential coefficients of the first and second order. Therefore we notice that, in order to determine the measure of curvature, it is necessary to know only the general expression for a linear element ; the expressions for the coordinates x, y, z are not required. A direct result from this is the remarkable theorem : If a curved surface, or a part...
Page 47 - ... shall be justified by pregnant theorems. The solution of the problem, to find the measure of curvature at any point of a curved surface, appears in different forms according to the manner in which the nature of the curved surface is given. When the points in space, in general, are distinguished by three rectangular coordinates, the simplest method is to express one coordinate as a function of the other two. In this way we obtain the simplest expression for the measure of curvature. But, at the...
Page 49 - By means of this theorem we obtain the angles of a plane triangle, correct to magnitudes of the fourth order, if we diminish each angle of the corresponding spherical triangle by one-third of the spherical excess. In the case of nonspherical surfaces, we must apply unequal reductions to the angles, and this inequality, generally speaking, is a magnitude of the third order. However, even if the whole surface...
Page 49 - However, even if the whole surface differs only a little from the spherical form, it will still involve also a factor denoting the degree of the deviation from the spherical form. It is unquestionably important for the higher geodesy that we be able to calculate the inequalities of those reductions and thereby obtain the thorough conviction that, for all measurable triangles on the surface of the earth, they are to be regarded as quite insensible.