COM-POS'ITE , adj. composite function . (1)
See COMPOSITION -composition of functions. (2) A function which is factorable (can be written as the product of two or more func tions), 
(usually refers only lo polynomial functions which are factorable relative to some specified field).
composite hypothesis . See HYPOTHESIS.
composite life of a plan t. The time required for the total annual depreciation charge to accumulate, at a given rate of interest, to the original wearing value.
composite number . A number that has two or more prime factors, as 4, 6, or 10, in distinc tion to 
and prime numbers such as 3, 5, or 7. Refers only to integers, not to rational or irrational numbers.
composite quantity . A quantity that is factorable.
derivative and differential of a composite function. See chain— chain rule, differ ential.
COM'PO-SI'TION , n. composition in a pro portion. Passing from the statement of the proportion to the statement that the sum of the first antecedent and its consequent is to its consequent as the sum of the second antecedent and its consequent is lo its consequent; i.e., passing from 
composition and division in a proportion. Passing from the statement of the proportion to the statement that the sum of the first antecedent and its consequent is to ihe difference between Ihe first antecedent and its consequent as the sum of the second antecedent and its consequent is to the difference between the second antecedent and its consequent; i.e., passing from 
to 
See above, composition in a proportion, and division division in a proportion. composition of functions . Forming a new function h (the composite function) from given functions g and f by the rule that 
for all x in the domain of f for which 
is in the domain of g. For example, if 
when 
and 
for
all real numbers, then the domain of the composite h of g and f is the set of all x with
In particular, 
This composite h of g
and f often is written as 
or 
but some times as 
or jg (as for the composition of relations, of which functions are speciaf eases). The order in which functions are combined is important. For example, if 
and 
t hen 
, but 
The derivative of the composite of two (functions can be computed by use of the chain rule,
composition of relations . Given a relation R and a relation S, the composite relation 
is the relation for which x is related to z if and only if there is an object y for which 
and 
For example, if r, s and t
denote positive integers, 
means 
and
means "r divides s," then 
means
"there is an integer s greater than r that divides t ," and 
means "there is a positive
integer s less than t that is divisible by r," See relation, and above, composition of functions.
composition of tensors . See inner inner product of tensors.
composition of vectors . The same process as addition of vectors, but the term com position of vectors is used more when speaking of adding vectors which denote forces, velocities, or accelerations; finding the vector which represents the resultant of forces, velocities, accelerations, etc., represented by the given vectors. See sum —sum of vectors.
graphing by composition of ordinates . See graphino —graphing by composition.
COM'POUND , adj. compound amount and compound interest . Sec INTEREST.
compound event . See EVENT—com pound event.
compound number . The sum of two or more denominations of a certain kind of denominate number; e.g., 5 feet, 7 inches, or 6 pounds, 3 ounces.
compound survivorship life insurance. See insurance life insurance.
COM-PRES'SION , n. See tension.
modulus of compression . See modulus — bulk modulus.
simple or one-dimensional compressions. Same as one-dimensional strains. See STRAIN.
COM'PU-TA'TION , n. The act of carrying out mathematical processes; used mostly with reference to arithmetic rather than algebraic work. One might say, "Find the formula for the number of gallons in a sphere of radius r and compute the result for r = 5"; or ''Com pute the square root of 3." Frequently used to designate long arithmetic or analytic proc esses that give numerical results, as computing the orbit of a planet.
compulation by logarithms. See logarithm.
numerical computation . A computation in volving numbers only, not letters representing numbers.
COMPUTE ', v. To make a computation. Syn. calculate.
COM-PUT'ER , n. Any instrument which performs numerical mathematical operations. A mechanical machine which primarily per forms combinations of addition, subtraction, multiplication and division is sometimes called a calculating machine in distinction from such more versatile instruments as electronic computers. Syn. computing machine.
analog computer . A computing machine in which numbers are converted into measurable quantities, such as lengths or voltages, that can be combined in accordance with the desired arithmetic operations; e.g., a slide rule. Generally, if two physical systems have corresponding behavior, and one is chosen for study in place of the other (because of greater familiarity, economy, feasibility, or other factors), then the first is called an analog device, analog machine, or analog computer. See BUSH.
digital computer . A computing machine that performs mathematical operations on numbers expressed by means of digits. Syn. digital device. See babbage, einiac, LEIBNIZ.
CON'CAVE, adj. c oncave toward a point (or
line or plane). A curve is concave toward a point (or line) if every segment of the arc cut off by a chord lies on the chord or on the opposite side of the chord from the point (or line). If there exists a horizontal line such that the curve lies above it and is concave toward it {lies below it and is concave toward it) the curve is said to be concave downward (upward). A curve is concave upward if and only if it is the graph of a convex function (see CONVEX—convex function). A circle with center on the x-axis is concave toward that axis, the upper half being concave down and the lower half concave up.
concave function . The negative of a convex function. See convex —convex function.
concave polygon and polyhedron . See POLYGON, POLYHEDRON.
CON-CAV'I-TY, n. The state or property of being concave.
CON'CEN-TRA'TION, n. concentration method for the potential of a complex . See POTENTIAL.
CON-CEN'TRIC , adj. concentric circles. Circles lying in the same plane and having a common center. Concentric is applied to any two figures which have centers (that is, are symmetric about some point) when their centers arc coincident. Concentric is opposed lo eccentric, meaning not concentric,
CON'CHOID, n. The locus of one end point of a segment of constant length, on a line which rotates about a fixed point ( 
in figure), the other end point of the segment being at the intersection 
of this line with a
fixed line not containing the fixed point. If the polar axis is taken through the fixed point and perpendicular to the fixed line, the length of" the segment is taken as It, and the distance from Ihe lixed point to the ti\cd line as a, the polar equation of the conchoid is 
Us Cartesian equation is
The curve is asymptotic to the fixed line in both directions, and on both sides of it. If the length of the line segment is greater than the perpendicular distance from the pole to the fixed line, the curve forms a loop with a node at the pole. If these two distances are equal, it forms a cusp at the pole. Syn. conchoid of Nicomcdes.
CON-CLU'SION, n. conclusion of a theorem.
The statement which follows (or is to be proved to follow) as a consequence of the hypothesis of the theorem. See implication.
CON-CORD'ANT-LY , adv. concordantly oriented. See MANIFOLD, simplex.
CON'CRETE , adj. concrete number. A num ber referring to specific objects or units, as 3 people, or 3 houses. The number and its references are denoted by concrete number.
CON-CUR'RENT, adj, Having a point in common. A collection of sets is concurrent if there is a point that belongs to each of the sets. For example, the medians of a triangle are concurrent; these three lines all contain the point 
of the distance from a vertex to the opposite side along a median. In space, the planes that contain the origin are concurrent (they are a bundle of planes). See CON SISTENCY—consistency of linear equations.
CON-CYC'LIC , adj. concyclic points . Points which lie on a common circle.
CON'DEN-SA'TION , adj. condensation point.
A point P is a condensation point of a set S if each neighborhood of P contains uncountably many points of S. See accumulation- accumulation point, COUNTABLE—count able set.
CON-DI'TION, n. A mathematical assump tion or truth that suffices lo assure the truth of a certain statement, or which must be true if this statement is true. A condition from which a given statement logically follows is a sufficient condition ; a condition which is a logical consequence of a given statement is a necessary condition . A necessary and suffi cient condition is a condition that is both necessary and sufficient. A condition may be necessary but not sufficient, or sufficient but not necessary. It is necessary that a sub stance be sweet in order that it be called sugar, but it may be sweet and be arsenic; it is sufficient that it be granulated and have the chemical properties of sugar, but it can be sugar without being granulated. In order for a quadrilateral lo be a parallelogram, it is necessary, but not sufficient , that two opposite sides be equal, and sullicient, although not necessary, that all of its sides be equal; but it is necessary and sufficient that two opposite sides be equal and parallel. Sec implication.
CON-DI'TION-AL , adj. conditional con vergence of series. See CONVERGENCE—con ditional convergence.
conditional equation and inequality . See EQUATION, INEQUALITY.
conditional probability. See probability.
conditional statement. Same as implication.
CON-DUC'TOR , adj., n. conductor potential. For a region R with boundary S, the con ductor potential is the function harmonic in the interior of R, continuous on 
and taking on the constant value 1 on S. It describes the potential of an electric charge in equilibrium on the surface of a conductor.
CONE , n. (1) A conical surface (sec conical conical surface). (2) A solid boun ded by a region (the base ) in a plane and the surface formed by the straight line segments (the elcments) which join points of the boundary of the base to a fixed point (the vertex) not in the plane of the base (the surface bounding this solid is also called a cone). The perpendicular distance from the vertex to the plane of the base is the altitude of the cone. If the base has a center, the line passing through the center of the base and the vertex is the axis of the cone. The cone is circular or elliptic in the cases its base is a circle or ellipse (some- limes a circular cone is defined to be a cone whose intersections with planes perpendicular t o the axis, but not intersecting the base, arc circles). An oblique circular cone is a circular cone whose axis is not perpendicular to its base. A right circular cone (or cone of rev olution ) is a circular cone whose base is perpendicular to its axis (sometimes called simply a circular cone). A right circular cone can be generated by revolving a right triangle about one of its legs, or an isosceles triangle about its altitude. The slant height of a right circular cone is the length of an element of the cone. The lateral are a of a cone is the area of the surface formed by the elements (for a right circular cone this is equal to 
where r is the radius of the base and 
is the slant height). The volume of a cone is equal to one-third of the product of the area of the base and the altitude. If the cone has a circular base, the volume is 
where r is the radius of the base and s is the altitude.
frustum of a cone. The part of the cone bounded by the base and a plane parallel to the base (sec figure). The volume of a frustum of a cone equals one-third the alti tude (the distance between the planes) times the sum of the areas of the bases and the
square root of the product of the areas of the bases; i.e.. 
The lateral area of a frustum of a right circular cone (the area of the curved surface) is equal to 
where / is the slant height and r and 
are the radii of the bases.
ruling of a cone . See RULING.
spherical cone . A surface composed of the spherical surface of a spherical segment and the conical surface defined by the bounding circle of the segment and the center of the sphere (see conical conical surface); a spherical sector whose curved base is a zone of one base. The volume of a spherical eone is 
where r is the radius of the sphere and h is the altitude of the zone base.
tangent cone of a quadric surface . A cone whose elements arc each tangent to the quadric. In particular, a tangent cone of a sphere is any circular cone all of whose elements are tangent to the sphere. If a ball is dropped into a circular cone, the cone is tangent to the ball.
truncated cone. The portion of a cone included between two nonparallcl planes whose line of intersection docs not pierce the cone. The two plane sections of the cone are the bases of the truncated cone.
CON'FI-DENCE , adj . confidence interval. An interval which is believed, with a preas- signed degree of confidence, to include the particular value of some parameter being estimated. Tech. For a random variable X whose distribution depends on an unknown parameter 
confidence interval is an interval 
for which 
and 
are statistics, i.e., functions of random sam ples 
and Prob 
If repeated samples are taken and 
and 
are computed for each sample, we would find, on the average, that at least the pro portion 
of the intervals did include 
and not more than the proportion 
did not.
For example, for a normal distribution with unknown mean 
known variance 
, and a random sample 
with
we have
and 
confidence interval if 
is chosen as 
so that, for a normal distribution with mean 
and variance 1, 
This can also be written as
If the variance also is unknown, the 
confidence interval is
where 
s is the .sample standard deviation.
is the 
percentile of the t -distribution with n - 1 degrees of freedom.
confidence region . For a random variable 
whose distribution depends on unknown parameters 
confi-
dence region is a set S in the re-dimensional space of possible values for 
that is determined in some way by sample values 
and satisfies 
most selective confidence interval . A 
confidence interval 
(see above;, confidence interval) which has the property that, if 
is any other 
confidence interval, then, for any incorrect value 
of the unknown param eter, we have
That is, the most selective confidence inter vals cover false values of the parameters with minimal probability. Sometimes Called SHORTEST CONFIDENCE INTERVAL, but the most selective confidence interval is not necessarily shortest in the sense of minimizing 
unbiased confidence interval . A confi dence interval 
(see above, confi dence interval) which has the property that the conditional probability of 
given that 
is the true value of the unknown parameter, is never less than the conditional probability of 
given that the parameter has some other value.
CON-FIG'U-RA'TTON , n. A general term for any geometrical figure, or any combination of geometrical elements, such as points, lines, curves, and surfaces.
CON-FO'CAL , adj. confocal conics . Conies having their foci coincident. E.g., the ellipses and hyperbolas represented by the equation
where 
and k lakes on all other real values for which 
arc confocal. These
conies intersect at right angles, forming an orthogonal system. (See point P in figure.)
confocal quadrics - Quadrics whose principal planes are the same and whose sections by any one of these planes are confocal conks. E.g., if k is a parameter and a, b, and c are fixed, the equation
represents a triply orthogonal system of confocal quadrics: For 
the equation represents a family of confocal ellipsoids ; for 
it repre-
sents a family of confocal hyperboloids of one sheet ; and for 
it represents
a family of confocal hyperboloids of two sheets . Each member of one family is confocal and orthogonal to each member of other families. See orthogonal- triply orthogonal system of surfaces. For 
we get, by a limiting process, the elliptic portion of the 
-plane (counted twice) bounded by (I) 
similarly for 
we get the hyperbolic portion of the 
-plane (counted twice) bounded by (2) 
Equations (1) and (2) define the focal ellipse and the focal hyperbola , respectively, of the system. Through each point 
of space there pass three quadrics of the system. The corresponding values 
of k are called the ellipsoidal coordinates of 
Sec coordinate — ellipsoidal coordinates.
CON-FORM'A-BLE, adj. conformable mat rices . Two matrices A and 11 such that the number of columns in A is equal to the number of rows in B. It is possible to form t he product AB if, and only if, A and B are conformable. Being conformable is not a symmetric; relation. See PRODUCT—product of matrices.
CON-FORM'AL, adj. conformal-conjugate representation of one surface on another. A representation which both is conformal and is such that each conjugate system on one surface corresponds to a conjugate system on the other. Syn. isothermal-conjugate repre sentation of one surface on another.
conformal map or conformal transformation. A map which preserves angles; i.e., a map such that if two curves intersect at an angle 
then the images of the two curves in the map also intersect at the same angle 
The functions 
map the 
-domain ol definition conformally on a surface S if and only if the funda mental quantities of the first order satisfy 
See isot he r MIC —isothermic map. The coordinates u, v arc called conformal parameters . The corres pondence between surfaces 
and 
deter mined by 
and 
is conformal at regular points if, and only if, the fundamental quantities of the first order satisfy 
The only conformal correspondences between open sets in three-dimensional Euclidean space are obtained by inversions in spheres, reflections in planes, translations, and magnifications. See cauchy —Cauchy-Riemann partial differ ential equations.
conformal parameters - See above, con formal map,
CON'GRU-ENCE, n. A statement of type 
which is read: "x is congruent to y modulus (or modulo) w"; w is the modulus of the congruence. When x, y, and w are integers, the congruence is equivalent to the statement "x — y is divisible by w," or "there is an integer k for which 
." E.g., 
(mod 7), since 
and 14 is divisible by 7. For a given positive integer n, a modular arithmetic or arithemetic modulo n is obtained by using only the integers 
and defining addition and multiplication by letting the sum 
and the product ab be the remainder after division by n of the ordinary sum and product of a and b. E.g., if 
then 
and the multiplicative inverse of 2 is 4, since 
If n = 15, then 3 has no multipli cative inverse, since a multiplicative inverse a would have the property that 
for some integer k and that 
Arithmetic modulo n is a commutative ring with unit clement; if n is a prime, then arith metic modulo n is afield [see FIELD, and ring]. Congruences may be used in many situations. If polynomials arc used, then with the modulus chosen as the polynomial 
the congruence 
means that 
is divisible by 
E.g., 
since 
If x and y are members of a group and W is a subgroup, then 
means that 
belongs to W. E.g., if x and y are complex numbers and W is the set of real numbers, then 
(mod W) could be defined as meaning that 
is a real number, or as meaning that 
is a real number. See fermat Format's theorem.
linear congruence. A congruence in which all terms are of the first degree in the variables involved. E.g., 
(mod 42) is a linear congruence.
quadratic congruence . A congruence of the second degree. Thus its general form is 
(mod n), where 
CON'GRU-ENT , adj. congruent figures . In plane geometry , it is customary to say that two figures are congruent if one of them can be made to coincide with the other by a rigid motion in space (i.e., by translations and rotations in space). Thus it might be said that two figures are congruent if they "differ only in location." Two line segments of equal length are congruent and two circles of equal radii are congruent. Each of the following is a necessary and sufficient condition for two triangles to be congruent: (i) There is a one-to-one correspondence between the sides of one triangle and the sides of the other for which corresponding sides arc equal (abbrev.: SSS); (ii) there is a one-to-one correspondence between the sides of one triangle and the sides of the other for which two sides and the angle determined by these sides arc equal, respec tively, to the corresponding sides of the other triangle and the angle determined by these sides (SAS); (iii) there is a one-to-one corres pondence between the angles of one triangle and the angles of the other for which twoangles and the side between the vertices of these angles are equal, respectively, to the corresponding angles of the other triangle and the side between the vertices of these angles (ASA). If we change the definition of congruence to allow only rigid motions in the plane, a different concept of congruence results. In solid geometry , two figures are congruent if one of them can be made to coincide with the other by a rigid motion in space. Sometimes such figures are said to be directly congruent and two figures for which one is directly congruent to the reflection of the other through a plane are oppositely congruent (then two figures are either directly or oppositely congruent if and only if one can he made 10 coincide with the other by a rigid motion in four-dimensional space). Often when giving axioms for a geometric system, congruence is taken as an undefined concept restricted by suitable axioms.
congruent matrices. See transformation — congruent transformation.
congruent numbers, or quantities . See CONGRUENCE.
congruent transformation . See TRANSFOR MATION— congruent transformation.