surf2 (serf)

[An [An altered form of suff 2 for sough2: see sough2. Cf. surf1 for sough1.] The bottom or conduit of a drain. Imp. Dict. surface (ser'fas), n. and a. [ The whole architecture of the house [in Pompeii] was coloured, but even this was not considered so important as the paintings which covered the flat surfaces of the walls. J. Fergu on, Hist. Arch., I. 370. 2. The boundary between two solid spaces not adjacent to a third: distinguished as a mat he matical surface. A surface is a geometrical locus defined by a single general and continuous condition. This condition reduces the points of the surface to a two dimensional continuum, its enveloping planes to a two dimensional continuum, and its enveloping straight lines to a three-dimensional continuum. A ruled surface appears to be enveloped by a one-dimensional series of lines; but when imaginary points are considered, this is seen not to be so. A true one-dimensional continuum of lines requires for its determination a threefold condition, and can contain but a finite number (or discrete infinity) of points and of planes. The number of points or planes of a surface which satisfy a twofold additional condition, as that the points shall lie upon a given line, or that the planes shall contain a given line, and the number of lines of the surface which satisfy a threefold additional condition, as that they shall belong to a given plane pencil, are either finite or only discrete infinity. In the former case the surface is said to be algebraical, in the latter transcendental. If the imaginary elements are taken into account, the numbers are constant whatever the special lines or pencils to which they refer may be. The number of points of an algebraical surface which lie upon a given straight line is called the order of the surface; the num tangent planes which contain a given line is called the cla of the surface; and the number of tangent lines which belong to a given plane pencil is called the rank of the surface. 3. Outward or external appearance; what appears on a slight view or without examination. If we look below the surface of controversy, we shall commonly find more agreement and le disagreement than we had expected. J. R. Seeley, Nat. Religion, p. 4. 4. In fort ., that part of the side which is terminated by the flank prolonged and the angle of the nearest bastion. - Adjunct surface, a surface applicable to another with corresponding elements orthogonal. The two surfaces are a ociated minimal surfaces. - Algebraic surface, a surface which is represented in analytical geometry by an algebraic equation. If imaginary parts of the locus are included, it is characterized by having a finite order, cla , and rank.Aly eid surface, a surface generated by the rotation That you may well perceive I have not wrong'd you, of the catenary about its base. It is the only surface of One of the greatest in the Christian world revolution for which the principal radii of curvature are Shall be my surety. Shak., All's Well, iv. 4. 3. everywhere equal and opposite, - An all agma tic, anti He that is surety for a stranger shall smart for it. Prov. xi. 15. surface clastic, apsidal surface. See the adjectives. - Apolar surface, a surface whose polar relatively to another surface (whose cla is at least as high as the order of the former) is indeterminate. - Applicable surface, a surface related to another surface in such a way that if they are brought in contact at any one point, and one is then rolled over the other so that a certain point P of the latter comes in contact with the other, then a variation of the path of the rolling will not in general cause a different point of the former surface to come into contact with the point P.- A ociated surface, a surface so applicable to another that corresponding elements make a constant angle with one another. The two surfaces are minimal surfaces having their tangent planes at corresponding points parallel. - Augmented surface. See aug ment. - Bonnet's surface, a minimal surface spherically represented by two families of circles, its equations being x = A COS a + sin cosh μ; y = - + COs a cos a sinh μ; z = sin a cos a cosh μ; where A and u are the parameters of the lines of curvature, and a is constant. Its section by the planes of XY shows an infinite series of equal catenaries having their bases parallel to Y. These are lines of curvature, and their planes cut the surface under the constant angle a.-Canal surface, a surface generated by a plane curve whose plane rolls upon a developable without slipping. Cen tral surface. (a) A surface having a center. (b) A centrosurface. Cla of a surface. See def. 2.- Closed surface. See close1. - Complex surface, a quartic surface having a nodal line and eight nodes. These lie on four planes through the nodal line, the section of the surface by each of these planes being a twofold line. The 6082 stant, where 8, p, q, r are independent linear functions of the coördinates, and where = 8 + p2 + q2 + ra + 2a ( sp + qr )+ 2b ( sq + pr )+2c(sr+ pq ). - Level surface. Same as equipotential surface (which see, under equipoten trial ). - Mean surface, the locus of the point midway between the points of tangency of lines of an isotropic congruence which are simultaneously tangent to two mutually applicable surfaces. - Minimal surface. (a) A surface within which lies an area the least po ible under given conditions. (b) An ela oidal surface (which see, above): an ordinary use, but not quite accurate. - Molding surface, a surface generated by a plane curve whose plane rolls upon a cylindrical surface. It is a species of canal surface. - Monoidal surface, a surface with a point having a degree of manifoldne one le than the order of the surface.-Neutral surface, a developable whose generators are the neutral axes of a beam.- Normopolar surface, the locus of the poles of a plane with reference to a given quadric surface-that plane containing three feet of normals from a variable point to that quadric.-Octadic surface. See octadic.-Orange skin surface. See or angel.-Order of an algebraic surface. See def. 2.- Parallel surfaces. See parallel curves, under parallel. - Pencil of surfaces. See pencill. -Plane surface, a surface in which if any two points are taken the straight line connecting them lies wholly in that surface. - Polar, popliteal, prone, pseudo spherical, quadric surface. See the adjectives. which cut any given line in the surface. Rank of a surface. See rank2.-Ray surface, a ruled surface generated by rays reflected or refracted at a skew curve. -Reciprocal surface, a surface every tangent plane of which is the polar of a point of a primitive surface rela a ruled surface, the number of generators to it through the nodal line belong to a complex of the second order. - Conical surface. See conical. - Contact of surfaces. See contact. - Counter pedal, cubic, cyclifying, cylindric surface. See the adjectives. - Cyclic surface, a surface generated by a circle varying in position and radius. - Cyclide surface. (a) A surface of the fourth order having the absolute circle as a nodal line. Sometimes distinguished as Darboux's cyclide. (b) A special case of the above, with four conical points. Generally distinguished as Dupin's cyclide. - Cyclotomic surface, a surface generated by a variable circle whose center is fixed, and which rotated round a fixed axis while constantly touching a fixed curve. - Developable surface, a surface that can be unwrapped in a plane without any doubling of parts over one another, or separation, as the surfaces of the cylinder and cone. See developable. -Diagonal surface, a special surface of the third order.-Dianodal, dorsal, equal, equipotential surface. See the adjectives. - Double surface, a surface the locus of the middle of chords of a minimal curve or imaginary curve every tangent of which touches the absolute circle. It is a minimal surface. - Doubly connected surface, a ring-shaped surface, one on which it is po ible to draw an oval so that a point may move from the outside to the inside without traversing the curve (more accurately speaking, the oval has no distinction of inside and outside); but after one such oval is drawn it is impo ible to draw another not intersecting the first. -Ela oidal surface, a surface whose mean curvature is nothing: same as minimal surface, in the sense in which the latter is commonly used. Enneper's surface [invented by A. Enneper in 1864], a surface of constant curvature, but not of revolution, of which one set of lines of curvature are plane or spherical.- Equatorial surface, a complex surface having its nodal line at infinity. - Eroded surface. See erode. Family of surfaces, in math., all the surfaces which are generated by a curvé of a general kind moving in a general way. - Flattened surface, a surface consisting of a multiple plane with nodal curves and points. - Focal surface, a surface having the lines of a primitive congruence as bit an gents. See Malus's theorem, under theorem. - Fresnel's surface of elasticity. See elasticity. - Gauche surface. See gauche. Generating surface. See generate. - Helicoidal surface, a surface generated by the helicoidal motion of a curve. All cylindrical surfaces and surfaces of revolution are helicoidal surfaces. - Henneberg's surface [invented by L. Henneberg in 1875], a double ela oidal surface of the fifth cla . He ian surface [named after Dr. Otto He e: see He ian2 ], the locus of points whose polar quadrics relatively to a primitive surface are cones. It cuts the primitive surface in the parabolic curve of the latter. Hyper cyclic surface, a surface belonging to one of two systems which form a We in gar ten i an triplet of constant flexure with a system of pseudo spherical surfaces. - Hyper jacob i an surface, a surface whose equation is formed by equating to zero a functional determinant formed of three columns of the Jacob i an matrix of three surfaces. See hyper jacob i an, -Inclined polar surface of a given pole in reference to a given primitive surface and for given angle, locus of a point whose polar plane in reference to the given primitive circle is inclined by the given angle to the line from the variable point to the pole. - Indicatrix surface, a quadric surface whose equation is IMG:content-2550.png:[blocks in formation] the former surface is also the pole of a tangent plane of the latter. - Rectifying developable surface of a non-plane curve. See rectify. Refracting surface. See refracting. - Respiratory surface. See respiratory. Riemann's surface [named from its inventor, the German mathematician G. F. B. Riemann (1826-66)], an imaginary surface to represent an n-valued function by n infinite planes cro ing into one another along certain lines, each of these planes representing the whole spread of imaginary quantity, and one value of the function belonging to each point of each plane. - Roman surface. Same as Steiner's surface (b). - Ruled surface. See rule1.-Screw surface. (a) A helicoidal surface. (b) A surface generated by the helicoidal motion of a right line. Self-reciprocal or sib i reciprocal surface, a surface whose reciprocal has the same order and singularities as itself. - Singly connected surface, a surface on which it is impo ible to pa from the inside to the outside of an oval or closed curve drawn on the surface without cro ing the surface. Skew surface. See skew1.-Spiral surface, a surface generated by a curve the plane of which rotates uniformly an axis in that plane, while the plane, and the curve with it, undergo expansion in a constant ratio per unit of time away from a center in the axis of rotation. - Steinerian surface, the locus of the vertices of cones which are polars quadrics of points with reference to a given primitive surface. - Steiner's surface. (a) A Steinerian surface. (b) The surface often originally, and better, called the Roman surface [discovered by Jacob Steiner (1796-1863), undoubtedly the greatest of all geometricians], being a quartic surface of the third cla , having three double lines. In its symmetrical form its appearance is thus described: Take a tetrahedron, and inscribe in each face a circle. There will be, of course, two circles touching at the mid-point of each edge of the tetrahedron; each circle will contain, on its circumference, at angular distances of 120°, three mid-points; and the lines joining these with the center of the tetrahedron, produced beyond the center, meet the opposite edges... joining the mid-points... Now truncate the tetrahedron by planes parallel to the faces, so as to reduce the altitudes, each to three fourths of the original value; and from the center of each new face round off symmetrically up to the adjacent three circles; and within each circle scoop down to the center of the tetrahedron, the bounding surface of the excavation pa ing through [that is, containing] the three right lines, and the sections by planes parallel to the face being in the neighborhood of the face nearly circular, but, as they approach the center, a uming a trigoidal form, and being close to the center an indefinitely small equilateral triangle. We have thus the surface, consisting of four lobes united only by the lines through the mid-points of opposite edges - these lines be ing consequently nodal lines, the mid-points being pinch points surface, and the faces singular planes, touching the surface along the inscribed circle. ( Cayley, Proceedings London Math. Soc., V. 14.)-Surface of aberration, the ruled surface described in a year by the line of apparent direction of a star as affected by aberration.-Surface of centers. See center1. - Surface of discontinuity, a vortex sheet within a fluid over which slipping takes place. - Surface of equal head. See head. Surface of revolution, a surface which is gen erated by the revolution of a curve round an axis. Sur face of translation. (a) A cylindrical surface. (b) More generally, a surface generated by a curve the plane of which moves in any way so that every line in it remains parallel to itself. - Synclastic surface, a surface which at each point has both its principal centers of curvature on the same side. - System of surfaces, a continuum of surfaces of a given order between the coördinates of whose point-equations a number of homogeneous equations subsist. Tabular surface, a surface generated by a given curve, and its plane at right angles to the tangent of that curve.-Tasimetric surface, a quadric surface such that when it is represented by the equation a circle of a given radius, which moves with its center on Ax + By2 + Cz2 + 2Dxy + 2Exz+ 2Fyz = 1, - Und eve lop able surface, a surface that cannot be developed in the plane: opposed to developable surface.Vicinal surface, a surface every point of which is infinitely near (but not equally near) another surface. = Syn. 1. Superficies, Exterior, etc. See outside. II. a. Of or pertaining to the surface; external; hence, superficial; specious; insincere: as, mere surface politene or loyalty. We were friends in that smooth surface way We Ru ians have imported out of France. T. B. Aldrich, Pauline Pavlovna. Surface condensation, paper, etc. See the nouns Surface right. See mineral right, under right. surface (sér fās), v. t.; pret. and pp. surfaced, ppr. surfacing. [ From Great Falls to Helena, been surfaced all the way. [the track] had not C. D. Warner, Harper's Mag., LXXVL 566. Surfaced paper. See paper . surface-car (ser'fās-kär), n. A car moving on rails laid on the surface of the ground, as distinguished from one moving on an elevated or an underground railway. [U. S.] "Come, now!" or "Now we're off!" are good starting commands, and the American isms one hears upon the front platforms of New-York surface cars should be carefully avoided. surface-chuck (ser'fās-chuk), n. A face-plate chuck in a lathe, to which an object is fixed for surface-color (sėr'fās-kulor), n. A color or pigment used in surface-printing. New York Tribune, May 11, 1890. turning.