## General Investigations of Curved Surfaces of 1827 and 1825 |

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abscissas arbitrary auxiliary sphere BDD'B C. F. Gauss centre circa superficies curvas circle convex coordinates correct to terms cos2 curved line curved surface denote Derivation of formula differential dp dp dp dp dq dq dp dq dq dq drawn dx dy easily seen equation evidently expression figure formed by shortest function Gauss geodesic geometric given Hence infinitely small values integral curvature intersection INVESTIGATIONS OF CURVED KARL FRIEDRICH GAUSS length linear element Liouville's reprint magnitude mean curvature means measure of curvature method multiplying normal notation obtain paper plane curve pole polygon positive or negative preceding article quantities radii radius of curvature regarded represents the direction respectively right angles shortest lines side sixth degree spherical angle straight lines suppose tangent tion total curvature translation variables Wangerin Whence

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Page 118 - Allgemeine Auflösung der Aufgabe, die Theile einer gegebenen Fläche auf einer anderen gegebenen Fläche so abzubilden , dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird.

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 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".95104, 4".95131.

Page 87 - The angle between two planes is equal to the angle between the great circles which represent their orientations, and is therefore also measured by the angle between the poles of the great circles. 4) If x, у, г ; x', y', z...

Page iv - GÉNÉRALES sur les surfaces courbes; par MC-F. Gauss. Traduites en français, suivies de notes et d'études sur divers points de la théorie des surfaces et sur certaine classe de courbes; par ME Roger, ingénieur des mines.

Page 45 - ... partly in need of some limitations or closer determinations, which must be omitted here. In researches in which an infinity of directions of straight lines in space is concerned, it is advantageous to represent these directions by means of those points upon a fixed sphere, which are the end points of the radii drawn parallel to the lines. The centre and the radius of this auxiliary sphere are here quite arbitrary. The radius may be taken equal to unity. This procedure agrees fundamentally with...

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 46 - The author designates as measure of curvature at 400 a point of the curved surface the value of the fraction whose denominator is the area of the infinitely small part of the curved surface at this point and whose numerator is the area of the corresponding part of the surface of the auxiliary sphere, or the integral curvature of that element. It is clear that, according to the idea of the author, integral curvature and measure of curvature in the case of curved surfaces are analogous to what, in...

Page 30 - The excess over 180° of the sum of the angles of a triangle formed by shortest lines on a concavo-concave...