surface 
clastic, apsidal surface. Sue the adjectives. Apolar 
surface, a surface whose polar relatively to another sur- 
face (whose class is at least as high as the order of the 
former) is indeterminate. Applicable surface, sur- 
face related to another surface in such a way that if they 
are brought in contact at any one point, :uid one is then 
rolled over the other so that a certain point P of the lat- 
ter comes in contact with the other, then a variation of 
the patli of the rolling will not in general cause a ditfer- 
cnt point of the former surface to come into contact with 
the point V.- -Associated surface, a surface so applica- 
ble to another that corresponding elements make a con- 
stant angle with one another. The two surfaces are min- 
imal surfaces having their tangent planes at correspond- 
ing points parallel. Augmented surface. See au<i- 
ment. Bonnet's surface, a minimal surface spherically 
represented by two families of circles, its equations being 
x = A cos a + sin A cosh n ; 
y = fi + COS a COS A M II h i. ; 
z = sin a cos A cosh / ; 
where A and n are the parameters of the lines of curvature, 
and o is constant. Its section by the planes of XV 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. Ca- 
nal 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. (6) A cen- 
trosurface.- Class of a surface. See def. 2. Closed 
surface. See cfosi. Complex surface, a quartic sur- 
face 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 
surface derives its name from the fact that all tangents 
to it through the nodal line belong to a complex of the 
second order. Conical surface. Sec conical. Contact 
of surfaces. See contact. Counterpedal, cubic, cyc- 
lifying, cylindric surface. See the adjectives. Cyc- 
lic 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. Gen- 
erally 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 sur- 
face, 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 or- 
der. Dianodal, dorsal, equal, equipotential sur- 
face. 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 con- 
nected surface, a ring-shaped surface, one on which it 
is possible 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 impossible to draw another not intersecting the first. 
Elassoidal 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 1804], a surface of constant 
curvature, but not of revolution, of which one set of lines 
of curvature are plane or spherical. Equatorial sur- 
face, a complex surface having its nodal line at infinity. 
Eroded surface. See erwle. Family of surfaces, 
in math., all the surfaces which are generated by a curve 
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 hav- 
ing the lines of a primitive congruence as bitangents. 
See Malm's theorem, under theorem. Fresnel's sur- 
face of elasticity. See elaaticit)/. Gaucne surface 
See gauche. Generating surface. See generate. Hel- 
icoidal surface, a surface generated by the helicoidal 
motion of a curve. All cylindrical surfaces and surfaces 
of revolution nKhelicoidal surfaces. Henneberg's sur- 
face [invented by L. Hennebery in 1875], a double elas- 
soidal surface of the fifth class. Hessian surface 
[named after Dr. Otto Hesse: see Hessian?}, the locus of 
points whose polar quadrics relatively to a primitive sur- 
face are cones. It cuts the primitive surface in the par- 
abolic curve of the latter. Hypercyclic surface a sur- 
face belonging to one of two systems which form a Wein- 
gartenian triplet of constant flexure with a system of 
pseudospherical surfaces. Hyperjacobiau surface, a 
surface whose equation is formed by equating to zero a 
functional determinant formed of three columns of the 
.Tacobian matrix of three surfaces. See hyperjacoUan. 
Inclined polar surface of a given pole in reference 
to a given primitive surface and for a given angle, the 
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 
6082 
stant, where s, p, q, r are independent linear functions 
of the coordinates, and where </> = #'-' , p- -f- q- r- 
+ -2a(*p l qr) + '2b(sq \-pr)-\-'2c(sr + pq). Level surf ace. 
Sana- as c'juipotential surface (which see, under equipoten- 
tial). Mean surface, the locus of the point midway be- 
tween the points of tangency of lines of an isotropic con- 
gruence which are simultaneously tangent to two mu- 
tually applicable surfaces. Minimal surface. () A 
surface within which lies tin area the least possible under 
given conditions, (b) An elassoidal surface (which see, 
above): an ordinary use, but not quite accurate. Mold- 
ing 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 manifoldness one less than the 
order of the surface. Neutral surface, a developable 
whose generators are the neutral axes of a beam. Nor- 
mopolar surface, the locus of the poles of a plane with 
reference to a given quadric surface that plane con- 
taining three feet of normals from a variable point to 
that quadric. Octadic surface. See octadic. Orange- 
skin surface. See orange^. Order of an algebraic 
surface. See def. 2. Parallel surfaces. See parallel 
curves, under parallel. Pencil of surfaces. Seepe;r#i. 
Plane surface, a surface in which if any two points 
are taken the straight line connecting them lies whol- 
ly in that surface. Polar, popliteal, prone, pseudo- 
/xd j/d zd\ a 
Ife r dS + TJ * 
(Ax)"' du.dx 
'doc 
: d 1 -^ 
d-<j> 
Ir.il// (Ay] 

AxISz djTdz' (dz)5' dz 
A<f) A<t> A<t> n i 
Ax' Ay' dz' 
)-' dz.dj/ ' 
-., _ 
Ax.Az Ay.Az (dz) 3 
where <(, = is a primitive surface.- Jacobian surface 
ic locus of points whose polar planes with regard to four 
surfaces meet in a point. See ./<ieoM<in2. Kummer's 
surface [invented by E. F. Rummer in 18641 a quartic 
surface having sixteen nodes. Its equation is *= = Ksonr 
where K = a* + b- -[- c' - lobe - 1, o, b, and c being con' 
which cut any given line in the surface. Rank of a 
surface. See raitifl. 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- 
tively to an assumed quadric surface. Every point of 
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. Itiewann (1820-66)], an 
imaginary surface to represent an n-valued function by 
n infinite planes crossing 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 p^>int of each plane. Roman sur- 
face. Same as Steiner's surface (b). Ruled surface. 
See ntlei. Screw surface. () A helicoidal surface. 
(b) A surface generated by the helicoidal motion of a 
right line. Self-reciprocal or sibireciprocal sur- 
face, a surface whose reciprocal has the same order and 
singularities as itself. Singly connected surface, a 
surface on which it is impossible to pass from the inside 
to the outside of an oval or closed curve drawn on the 
surface without crossing the surface. Skew surface. 
See skewi. 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, under- 
go expansion in a constant ratio per unit of time away 
surface-motion 
Undevelopable surface, a surface that cannot be de- 
veloped in the plane: opposed to developable surface. 
Vicinal surface, a surface every point of which is infi- 
nitely near (but not equally near) another surface. = Syn. 
1. Superficies, Exterior, etc. See o-utside. 
II. a. Of or pertaining to the surface; exter- 
nal; hence, superficial; specious; insincere: 
as, mere surface politeness or loyalty. 
We were friends in that smooth surface way 
We Russians 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 (ser'fas), V. t. pret. and pp. mirl'iiml. 
ppr. furfaotng. [< surface, w.] To put a sur- 
face (of a particular kind) on, or give a (cer- 
tain) surface to ; specifically, to give a fine or 
even surface to ; make plain or smooth. 
From Great Falls to Helena, . . . [the track ] had not 
been surfaced all the way. 
C. D. Warner, Harper's Mag., LXXVI. 58. 
Surfaced paper. See paper. 
surface-car (ser'fas-kiir), n. A car moving on 
rails laid on the surface of the ground, as dis- 
tinguished from one moving on an elevated or 
an underground railway. [U. S.] 
" Com'e, now ! " or "Now we're off!"are good starting 
commands, and the Americanisms one hears upon the 
front platforms of New- York surface cars should be care- 
fully avoided. Xew York Tribune, May 11, 1890. 
surface-chuck (ser'fas-chuk), n. A face-plate 
chuck in a lathe, to which an object is fixed for 
turning. 
surface-color (ser'fas-kul"or), n. A color or 
pigment used in surface-printing. 
surface-condenser (ser'fas-kon-den' ( 'ser),i. 1. 
In steant-engin., a condenser iii which exhaust- 
steam is condensed by contact with surfaces of 
metal cooled by a flow of cold water on their 
sides opposite the condensing surfaces. .Such 
condensers are of various forms, those principally used 
for marine service consisting of a large number of small 
brass tubes inserted at their opposite ends in the sides of 
steam-tight chambers, and inclosed in a compartment 
through which cold sea-water is constantly forced by the 
circulating pump. The exhaust-steam enters one of the 
chambers, and on its passage through the tubes to the 
other chamber is condensed. The condensed water is 
continuously pumped back into the boilers. 
2. A metallic cone, or a series of pipes, heated 
quadrics of points with reference to a given primitive 
surface. Steiner's surface, (n) A Steinerian surface. 
(&) The surface often originally, and better, called the 
Roman surface [discovered by Jacob Sterner (1796-1868), 
undoubtedly the greatest of all geometricians], being a 
quartic surface of the third class, 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 dis- 
tances of 1'20, three mid-points; and the lines joining 
these with the center of the tetrahedron, produced be- 
parallel tc the faces, so as to reduce the altitudes, each 
to three fourths of the original value; and from the cen- 
ter 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 passing 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, assuming a 
trigoidal form, and being close to the center an indefi- 
nitely small equilateral triangle. We have thus the sur- 
face, 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 of the surface, and the faces singular planes, each 
touching the surface along the inscribed circle. (Cayley, 
Mah ' S c " v - 1 4 -) T Surface of ab- 
tion. Surface of centers. See center^. 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 coordinates of 
whose point-equations a number of homogeneous equa- 
tions subsist.- Tabular surface, a surface generated by 
a circle of a given radius, which moves with its center on 
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* 2 + Bi/- + Cz 2 -;- SQxy + 2Ezz ; ZVyz = 1, 
the coefficients are proportional to the components of a 
stress. Thlipsimetric surface, the same as a tasimet- 
ric surface, except that it represents a strain instead of a 
stress. Transcendental surface, a surface which is 
represented in analytical geometry by a transcendental 
equation. Tubular surface, the envelop of spheres of 
constant radius having their centers on a primitive curve. 
in a thin film to cause it to part 
by evaporation. See ei'ajiorating-cone. 
surfaced (ser'fast), . [< surface + -eiP.~\ 1. 
Having a surface of a specified kind, especially 
a fine surface ; made smooth. 
A profound delight in the beauty of the universe and in 
that delicately surfaced nature of his [Spenser's] which 
was its mirror and counterpart. 
Lowell, Among my Books, 2d ser., p. 187. 
2. Specifically, noting paper or cardboard that 
has received an additional thin coating or sur- 
face of filling to prepare it for a fine, sharp im- 
pression. 
surface-enamel (er'fas-e-nam"el), n. See en- 
amel, 3. 
surface-fish (ser'fas-fish), n. See fish 1 . 
surface-gage (ser''fas-gaj), . An instrument 
for testing the accuracy of plane surfaces. 
surface-geology (ser'fas-je-or'6-ji), . That 
branch of geological science which has to do 
with the distribution of the superficial or detri- 
tal formations, including also glacial geology, 
and the study of those erosive agencies which 
have given the earth's surface its present form. 
[Little used.] 
surface-glaze (ser'fas-glaz), n. Inceram., glaze 
which is thin and wholly transparent, and cov- 
ers the body and the decoration thinly. 
surface-grub (ser'fas-grub), M. The larva of 
any one of many different noctuid moths; a 
cutworm. Also surface-worm. 
surface-integral (ser'fas-in'te-gral), n. See in- 
tegral. 
surface-joint (ser'fas-joint), it. A joint which 
unites the margins of metallic sheets or plates. 
Such joints are generally formed by means of 
laps or flanges, soldered or riveted. E. H. 
Surfaceman (ser'fas-man), n. ; pi. surfacemen 
(-men). In rail., a person engaged in keeping 
the permanent way in order. [Eng.] 
surface-mining (ser'fas-mi /! 'ning), . Shallow 
mining, or that carried on at an inconsiderable 
depth beneath the surface; placer-mining, as 
generally denominated in California. Under this 
head A. J. Bowie (" Hydraulic Mining in California," p. 79) 
includes the methods of dry-washing, beach-mining, river- 
or bar-mining, ground-sluicing, and booming. 
Surface-motion (sei-'fas-m6"shon), . Motion 
at the surface. 
