CON'IC , n. Any curve which is the locus of a point which moves so that the ratio of its distance from a fixed point to its distance from a fixed line is constant. The ratio is the eccentricity of the curve , the fixed point the focus , and the fixed line the directrix . The eccentricity is always denoted by e. When e = 1, the conic is a parabola; when e < 1,
an ellipse; and when e > 1, a hyperbola. These are called conies, or conic sections , since they can always be gotten by taken plane sections of a conical surface (see dandelin). The equation of a conic can be given in various forms. E.g.; (I) If the eccentricity is e, the focus is at the pole, and the directrix perpen dicular lo the polar axis and at a distance q from the pole, the equation in polar coordi nates is 
This is equivalent to the following equation in Cartesian coordinates (the focus being at the origin and the directrix perpendicular to the x -axis at a distance q from the focus:
(2) The general algebraic equation of the
second degree in two variables always rep
resents a conic (including here degenerate
conies); i.e., an ellipse, hyperbola, parabola, a
straight line, a pair of straight lines, or a
point, provided it is satisfied by any real
points. See discriminant— discriminant
of a quadratic equation in two variables.
(3) See ELLIPSE, HYPERBOLA, PARABOLA.
acoustical, optical, or focal property of conics.
See ellipse —focal property of ellipse, hyphr- bola —focal property of hyperbola, and para bola —focal property of parabola.
central conics . Conies which have centers— ellipses and hyperbolas. See CENTER.
confocal conics . Sec confocal.
degenerate conic . A point, a straight line, or a pair of straight lines, which is a limiting form of a conic. E.g., the parabola ap proaches a straight line, counted twice, as the plane, whose intersection with a conical surface defines the parabola, moves into a position in which it contains a single element of the conical surface, and the parabola approaches a pair of parallel lines as the vertex of the cone recedes infinitely far; the ellipse becomes a point when the cutting plane passes through the vertex of the cone but does not contain an element; the hyperbola becomes a pair of intersecting lines when the cutting plane contains the vertex of the conical surface. All these limiting cases can be obtained algebraic ally by variation of the parameters in their several equations. See DISCRIMINANT—dis criminant of the general quadratic.
diameter of a conic . Sec diameter.
focal chords of conics. See focal.
similarly placed conics . Conics of the same type (both ellipses, both hyperbolas, or both parabolas) which have their corresponding axes parallel.
tangent to a general conic . (1) If the equation of the conic in Cartesian coordinates is 
then the equation of the tangent at the point (x1, y1) is
(2) If the equation of the conic in homo geneous Cartesian coordinates is written
then the equation of the tangent at the point 
is 
See coordinate —homogeneous coordinates,
CON'I-CAL , adj. conical surface . A surface which is the union of all lines that pass through a fixed point and intersect a fixed curve. The fixed point is the vertex , or apex , of the clinical surface, the curve the directrix , and each of the lines is a generator or genera trix. Any homogeneous equation in rect angular Cartesian coordinates is the equation of a conical surface with vertex at the origin.
circular conical surface . A conical surface whose directrix is a circle and whose vertex is on the line perpendicular to the plane of the circle and passing through the center of the circle. If the vertex is at the origin and the directrix in a plane perpendicular to the z-axis, its equation in rectangular Cartesian coordinates is 
quadric conical surfaces . Conical surfaces whose directrices arc conies,
CON'I-COID, n. An ellipsoid, hyperboloid, or paraboloid; usually docs not refer to limiting (degenerate) cases.
CON'JU-GATE , adj. complex conjugate of a matrix. Sec matrix —complex conjugate of a matrix.
conjugate algebraic numbers. Any set of numbers that are roots of the same irreducible algebraic equation with rational coefficients, an equation of the form:
E.g., the roots of 
which are 
and 
are conjugate algebraic numbers (in this case conjugate imaginary numbers).
conjugate angles . See angle— conjugate angles.
conjugate arcs . Two nonoverlapping arcs whose union is a complete circle.
conjugate axis of a hyperbola . See hyper bola.
conjugate complex numbers . See complex conjugate complex numbers.
conjugate convex functions. Sec convex — conjugate convex functions.
conjugate curves . Two curves each of which is a Bertrand curve with respect to the other. The only curves having more than one conjugate arc plane curves and circular helices.
conjugate diameters . A diameter and the diameter which occurs among the parallel chords that define the given diameter. The conjugate diameters in a circle arc perpendic ular. The axes of an ellipse are conjugate diameters. But, in general, conjugate diam eters are not perpendicular. See DIAMETER -diameter of a conic,
conjugate diameter of a diametral plane of a central quadric. The diameter which con tains the centers of all sections of the central quadric by planes parallel to a given diameter. The diametral plane is likewise said to be conjugate to the diameter.
conjugate directions on a surface at a point. The directions of a pair of conjugate diameters of the Dtipin indicatrix at an elliptic or hyper bolic point P of a surface S. There is a unique direction conjugate to any given direction on S through P, so that there are infinitely many pairs of conjugate directions on S at P. Two conjugate directions which are mutually perpendicular are necessarily principal directions. Conjugate directions are not defined at parabolic or planar points. The characteristic of the tangent plane to S, as the point of contact moves along a curve C on S, is the tangent to the surface in the direction conjugate to the direction of C. See below, conjugate system of curves on a surface.
conjugate dyads and dyadics. Sec DYAD .
conjugate elements and conjugate subgroups of a group. See transform transform of an element of a group.
conjugate elements of a determinant. Elements which are interchanged if the rows and columns of the determinant are interchanged; e.g., the element in the second row and third column is the conjugate of the 
element in the third row and second column, in general, the elements 
and 
are con jugate elements, 
being Ihe element in the i th row and i th column and 
the element in the 
row and i th column. See deter-
minant.
conjugate harmonic functions . See HAR MONIC—harmonic functions.
conjugate hyperboloids . Hyperboloids which with suitable choice of coordinate axes have equations
and 
Any plane containing the common axis 
cuts the two hyperboloids in conjugate hyper bolas. See hyperboloid —hyperboloid of one sheet, hyperboloid of two sheets.
conjugate imaginaries. See complex —con- jugate complex numbers.
conjugate points relative to a conic . Two points such that one of them lies on the line joining the points of contact of the two tan gents drawn to the conic from the other; two points that arc harmonic conjugates of the two points of intersection of the conic and the line drawn through the points; a point and any point on the polar of the point. Tech. If the conic is written in Ihe form
are homogeneous rectangular Carresian coordinates and 
then two points, 
and 
arc conjugate points if, and only if,
See below, harmonic conjugates wilh respect to two points.
conjugate quaternion s. See quaternion.
conjugate radicals. (1} Conjugate binomial surds (see surd). (2) Radicals that arc conjugate algebraic numbers.
conjugate roots . See above, conjugate al gebraic numbers,
conjugate ruled surfaces . See ruled.
conjugate space. If V is a vector space with scalars in a field F, then the
conjugate space of V is the vector space V* whose members are linear functions (functionals) with domain V and range contained in F. If V is finite- dimensional, then V and V* have the same dimension and V is isomorphic to its second conjugate space 
, the member.of 
corresponding to the member x of V being defined by 
for all/in 
If N is a normed vector space (real or complex) and f is a continuous linear function (function al) with domain N and range in the set of scalars (real or complex), then there is a least number, called the norm of f and written 
, such that 
for each x of N.
The set of all such functionals is a complete normed linear space, or lianach space, which is the first conjugate space of N, The first conjugate space of this space is the second con jugate space of N , etc. If N is finite-dimensional, then N and its second conjugate space are identical (i.e., isometric). For any normed linear space N, N is isometric with a subspace of its second conjugate space (see reflexive — reflexive Banach space). If N ' is a Hilbert space with a complete orthonormal sequence 
then the sequence of functions 
is a complete orthonormal sequence in the first conjugate space and the correspondence 
is an isometric correspondence between the 
two spaces. Syn. adjoint space; dual space.
conjugate subgroups. See isomorphism,
conjugate system of curves on a surface. Two one-parametcr families of curves on a surface S such that through each point P of S there passes a unique curve of each family, and such that the directions of the tangents to these two curves at P are conjugate direc tions on S at P. See conjugate- -conjugate directions on a surface at a point. The pa rametric curves form a conjugate system if, and only if, 
on S. See surface —funda-
mental coefficients of a surface. The lines of curvature form a conjugate system, and the only orthogonal conjugate system.
harmonic conjugates with respect to two points. Any two points that separate the line through the two points internally and extern ally in the same numerical ratio; two points (the 3rd and 4th) which with the given two {the 1st and 2nd) have a cross ratio equal to — 1 (see ratio). If two points are harmonic conjugates with respect to two others, the latter two are harmonic conjugates with re spect to the first two.
isogonal conjugate lines . See ISOGONAL— isogonal lines.
mean-conjugate curve on a surface . A curve C on a surface S such that C is tangent to a mean-conjugate direction on S at each point of C. See below, mean-conjugate directions on a surface.
mean-conjugate directions on a surface. Conjugate directions on the surface S at the point P of S such that the directions make equal angles with the lines of curvature of S at P. The mean-conjugate directions are real if the Gaussian curvature of S is positive at P, and the radius of normal curvature R of S in each of these two directions is the mean of the principal radii there: 
See above, mean-conjugate curves on a surface.
method of conjugate directions. A generalization of the method of conjugate gradients tor solving a system of n linear equations in n unknowns. In the method of conjugate directions, special restrictions on the con jugate directions to be used do not need to be specified.
method of conjugate gradients. An iterative method, terminating in n steps if there is no round-off error, for solving a svstem of n equations in n unknowns, 
Starling with an initial estimate 
of the solution vector x, the correction steps are in directions that are conjugate to each other relative to the matrix of coefficients, and (to within this constraint) they are successively chosen to be in gradient directions relative to an associated quadratic function that assumes its minimum value 0 at the solution x of the original problem. The sets of residuals are mutually orthogonal. See conjugate —con jugate points relative to a conic.
method of successive conjugates . In complex variable theory, an iterative method for the approximate evaluation of an analytic function that maps a given nearly circular domain con- formally onto the interior of a circle. This mapping might be considered as the second step in a two-step process of mapping a given simply connected domain conformally on the interior of a circle, the mapping of the given domain onto a nearly circular domain having previously been attained through known functions or through a catalogue of conformal maps.
CON-JUNC'TION n . conjunction of pro positions. The proposition formed from two given propositions by connecting them with the word and. E.g., the conjunction of "Today is Wednesday" and "My name is Harry" is the proposition "Today is Wednes day and my name is Harry." The conjunction of propositions p and q is usually written as 
or 
and read "p and q." The con-
junction of p and q is true if and only if both p and q are true. See disjunction.
CON-JUNCTIVE, adj. conjunctive trans formation. See TRANSFORMATION - conjunctive transformation.
CON-NECT' ED, adj . arcwise connected set. A set such that each pair of its points can be joined by a curve all of whose points are in the set. Syn. path connected set, pathwise connected set.
connected set of points. A set that cannot be separated into two sets U and V which have no points in common and which are such that no accumulation point of U belongs to V and no accumulation point of V belongs to U (see disconnected— disconnected set). The set of all rational numbers is not connected, since the set of rational numbers less than 
and the set greater than 
are both closed in the set of all rational numbers. An arcwise con nected set is connected, but a connected set need not be either arcwise connected or simply connected.
locally connected set . A set S such that, for any point x of S and neighborhood U of jit, there is a neighborhood V of x such that the intersection of S and V is connected and contained in U,
simply connected set . An aTcwise con nected set such that any closed curve within it can be deformed continuously to a point of the set without leaving the set. For a plane set, this means the set is arcwise con nected and no closed curve lying entirely within the set encloses a boundary point of the set. An arcwise connected set that is not simply connected is multiply con nected. See CONNECTIVITY- connectivity number.
CON'NEC-TIV'l-TY , adj. connectivity num ber. The connectivity number of a curve is 1 plus the maximum number of points that can be deleted without separating the curve into more than one piece (this is 
where
is the Euler characteristic). The connectivity number of a (connected) surface is 1 plus the largest number of closed cuts (or cuts joining points of previous cuts, or joining points of the boundary or a point of the boundary to a point of a previous cut, if the surface is not closed) which can be made without separating the surface. This is equal to 
for a closed surface and to 
for a surface with boundary curves. A simply connected curve or surface then has connec tivity number 1; a curve or surface is doubly connected, triply connected, etc., according as its connectivity number is 2, 3, etc. The region between two concentric circles in a plane is doubly connected; the surface of a doughnut (a torus) is triply con nected. In the above sense, the connectivity number of a connected simplicial complex (which may be a curve or a surface) is 1 plus the [-dimensional Betti number (modulo 2), However, the connectivity number is some times defined to be equal to this Belli number,
CO'NOID, n. (1) A surface that is the union of all straight lines parallel to a given plane, intersecting a given line, and intersecting a given curve. (2) A paraboloid of revolution, a hypcrboloid of revolution, or an ellipsoid of revolution. (3) The general paraboloid and hypcrboloid, but not the general ellipsoid.
right conoid . A conoid for which the given plane and the given line are mutually orthogonal.
CON-SEC'U-TTVE, adj. Following in order without jumping. E.g., in the sum 
the terms x and 
are consecutive terms; the sets 
and 
are sets of
consecutive integers ; and 
is a set of
consecutive odd integers . Such a concept can not be applied to the rational numbers, since for no rational number x is there a first rational number larger than x. Syn, successive,
consecutive angles and sides. Two consecutive angles of a polygon are two angles with a common side; two consecutive sides are two sides with a common vertex.
CON'SE-QUENT, n. (1) The second term of a ratio; the quantity to which the first term is compared, i.e., the divisor. E.g., in the ratio 
3 is the consequent, and 2 the antecedent. (2) See im plication.
CON'SER-VA'TION, n. conservation of energy. See lnlrqy.
CON-SER'VA-TIVE , adj. conservative field of force. A force field such that the work done in displacing a particle from one position to another is independent of the path along which the particle is displaced. In a conservative field the work done in moving a particle around any closed path is zero. IF the work done on the panicle is represented by a line integral, 
where 
and 
are the Cartesian components of force in a conservative field, then the integrand is an exact differential . The gravitational and electrostatic fields of force are examples of conservative fields, whereas the magnetic field due to current flowing in a wire and fields of force involving frictional effects are nonconscrvative.
CON-SIGN', v. to consign goods, or any property. To send it to someone to sell, usually at a fixed fee, in contrast to selling on commission.
CON'S1GN-EE' , n. A person to whom goods are consigned.
CON-SIGN'OR , n. A person who sends goods to another for him to sell; a person who consigns goods.
CON-SIST'EN-CY, n. consistency of systems of equations. The property possessed by a system of equations when there is at least one set of values of the variables that satisfies each equation, i.e., the solution sets have one or more common points. II' they are not satisfied by any one set of values of the variables, they are inconsistent . E.g., the equations 
and 
are in consistent; the equations 
and 
ire consistent, but are not inde pendent (see indepf.ndf.nt); and the equations 
and 
are consistent and independent. The first pair of equations represents two parallel lines, the second repre sents two coincident lines, and the third represents two distinct lines intersecting in a point, the point whose coordinates arc (3, 1). consistency of linear equations. A linear equation in two variables is the equation of a line in the plane. Therefore a single equation has an unlimited number of solutions. Two equations have a unique simultaneous solution if the lines they represent intersect and are not coincident; there is no solution if the lines are parallel and not coincident; there is an unlimited number of solutions if the lines are coincident. These correspond to the three cases of the following discussion. Consider the equations; 
where at least one of a iy bi and at least one of n 2 , bi is not zero. Multiply the first equation by b2 and the second by b\, then subtract. This gives 
Similarly. 
or 
and 
Three cases follow: I. If the determinant of the coefficients
is not zero, one can divide by it and secure unique values for x and y . The equations are then consistent and independent. The equa tions 
and 
reduce in the above way to 
and have the unique simultaneous solution 
II. Jf the determinant of the coefficients is zero and one of the determinants formed by replacing the coefficients of x (or of y) by the constant terms is not zero, there is no solution; i.e., the equations arc inconsistent. The equation 
reduceto 
which have no solution. III. If all three delerminants entering arc zero, there results 
and 
The equations arc then consistent but not independent. This is the situation for the equations 
and 
An infinite number of pairs of values of x and. y can be found that satisfy both of these equations. A linear equation in three variables is the equation of a plane in space. Therefore a single equation has an unlimited number of solutions. Two equa tions either represent parallel planes and have no common solution or else represent planes which intersect in a line or coincide and the equations have an unlimited number of solutions, Eliminating the variables, two at a time, from the equations
gives 
and 
where 
and 
are the determinants resulting from substituting the 
in the determinant of the coefficients, D, in place of the 
and 
respectively. Three cases arise: I. If 
it can be divided out and a unique set of values for x, y, and z obtained; i.e., the three planes representing the three equations then intersect at a point and the equations arc consistent (and also independent). II. If 
and at least one of 
and 
is not zero, there is no solution; the three planes do not have any paint in common and the three equations are inconsistent. III. If 
and 
three cases arise: a). Some second-order determinant in D is not zero, in which case the equations have infinitely many points in common; the planes (the loci of the equations) interect in a line and the equations are consistent, b). Every second-order minor in D is zero and a second- order minor in 
is not zero. The planes are then parallel but at least one pair do not coincide; the equations arc inconsistent.
r). All the second-order minors in 
and 
arc zero. The three planes then coincide and the equations are consistent (but not independent). The general situation of m linear equations in n unknowns is best handled by consideration of matrix rank (see matrix — rank of a matrix): the equations are consistent if and only if the rank of the matrix of the coefficients is equal to the rank of the aug mented matrix. If the constant terms in a system of linear equations are all zero (the equations are homogeneous), then the equa tions have a trivial solution (each variable equal to zero). For n homogeneous linear equations in m unknowns : (1) 
the equations have a nontrivial solution (not all variables zero). (2) 
the equations have a nontrivial solution if, and only if, the determinant of the coefficients is equal to zero. (3) 
m, the equations have a nontrivial solution if, and only if, the rank of the matrix of the coefficients is less than m. These arc simply the special ease of the results for n linear equations in m variables when the constant terms are all zero.
CON-SIST'ENT , adj. consistent assumptions, hypotheses, postulates. Assumptions, hypo theses, postulates that do not contradict each other.
consistent estimator (Statistics). An esti mator Ô for a parameter ô such that, for each positive number e. the probability that 
approaches 
as 
If an estimator is asymptotically unbiased, i.e., the expected value of 
approaches 
as 
and if the variance of 
ap- proaches 
then 
is a consistent estimator for $. 
is a random sample from a normal population and
where 
then 
which approaches 
as 
and the variance of 
is 
which approaches 
as 
Thus Ô is a consistent estimator for 
. See UNBIASED- unbiased estimator, variance.
consistent system of equations. See CONSISTENCY.
CON'STANT , n. A particular object or number. A symbol that represents the same object throughout a certain discussion or sequence of mathematical operations. A variable that can assume only one value. Sec VARIABLE.
absolute constant. A constant that never changes in value, such as numbers in arith metic.
arbitrary constant . A symbol thai stands for an unspecified constant. E.g., in the quadratic: equation 
the symbols 
and c are arbitrary constants. See below, constant of integration.
constant function . A function whose range has only one member; a function/for which there is an object a such that 
for all x in the domain of f.
constant of integration . An arbitrary con stant that must be added to any function arising from integration to obtain all the anti- derivatives. The integral, 
can have any of the values 
where c is a constant, because the "derivative of a constant is zero; further, it follows from the mean value theorem that there arc no other values for the integral. See mean— mean value theorems for deriva tives.
constant of proportionality or variation . See variation —direct variation.
constant speed and velocity. An object is said to have constant speed if it passes over equal distances in equal intervals of time (although the object need not move in a straight line). It has constant velocity if it passes over equal distances in the same direction in equal intervals of time (this means that the instantaneous velocity is the same vector at each point of the path; see velocity). Constant velocity is also sometimes called uniform (rectilinear) velocity and uniform motion (although uniform motion is some times used in such a sense as uniform circular motion, meaning motion around a circle with constant speed).
constant term in an equation or function . A term which docs not contain a variable. Syrt, absolute term.
essential constant. A set of essential con stants in an equation is a set of arbitrary constants which: (I) cannot be replaced by a smaller number (changing the form of the equation if desired) so as to have a new equation which represents essentially the same Family of curves, or (2) arc equal in number to the number of points needed to determine a unique member of the family of curves represented by the equation, or (3) are arbitrary constants in an equation 
for which the number of arbitrary constants is equal to the minimum order of a differential equation which has 
as a solution.
The linear equation 
defines a family of straight lines; it has 2 essential constants, since 2 points (not in a vertical line) determine a unique line of the family, and 
is a solution of the differential equation 
which is of order 2. The equation 
does not have 3 essential constants, since 2 points determine a line of the family of lines it represents; also, it represents the same family of curves as 
except for the lines x = constant. The equation 
has 4 arbitrary constants; these are not essen tial, since the equation can be factored, as 
and has the same family of curves as the equation 
The number of essential constants in an equa tion is the number of essential constants to which the arbitrary constants can be reduced. E.g., the number of essential constants in 
is 2. The constants 
in the equation 
are essential if and only if the functions 
are linearly independent,
gravitational constant. Sec gravitation law of universal gravitation.
Lame's constants . See LAM6—Lame's constants.