C.G.S. UNITS. Units of the centimeter — gram second system. Centimeter is the unit of distance (length); gram, mass; and second, time. See ERG, force -unit of force.
CABLE , n. parabolic cable. See parabolic.
CAL'CU-LATE, v . To carry out some mathematical process; to supply theory or formula and secure the results (numerical or otherwise) that are required; a looser and less technical term than compute. One may say, "Calculate the volume of a cylinder with radius 4' and altitude 5'"; he may also say. "Calculate the derivative of sin (2 x + 6)." Syn . compute.
CALCULATING MACHINE . Same as CUMPUTING MACHINE, but See COMPUTER.
CAL'CU-LUS , n. The field of mathematics which deals with differentiation and integra tion of functions and related concepts and applications. Sometimes called the infinitesi mal calculus because of the prominence of the use of infinitesimal in the early development of the subject. See below, differential calculus, integral calculus.
calculus of variations . The study of the theory of maxima and minima of definite integrals whose integrand is a known function of one or more independent variables and of one or more dependent variables and their derivatives, the problem being to determine the dependent variables so that the integral will be a maximum or a minimum. The simplest such integral is of the form
where y is to be determined to make I a maximum or a minimum (whichever is desired). The name calculus of variations originated as a result of notations introduced by Lagrange in about 1760 (see variation). Other integrals studied are of the form
where 
are unknown functions of x, or multiple integrals such as
where z is an unknown function of x and y, or multiple integrals of higher order or of various numbers of dependent variables (the integrand may also be a function of derivatives of higher order than the first). See brach-ISTOCHRONE — braehistochroneproblem, ISOPERIMETRIC -isoperimetric problem in the calculus of variations, euler Euler's equation.
differential calculus . The study of the variation of a function with respect to changes in the independent variable, or variables, by means of the concepts of derivative and differential; in particular, the study of slopes of curves, nonuniform velocities, accelerations, forces, approximations to the values of a function, maximum and minimum values of quantities, etc. See DERIVATIVE.
fundamental lemma of the calculus of variations. See fundamental.
fundamental theorem of calculus . See fun damental — fundamental theorem of calculus.
integral calculus. The study of integration as such and its application to finding areas, volumes, centroids, equations of curves, solutions of differential equations, etc.
CALL'A-BLE , adj. callable bond s. See bond.
CAL'O-RIE (or CALO-RY) , n. The amount of heat required to raise the temperature of one gram of waler one degree Centigrade. The calorie thus defined varies slightly for different temperatures. A standard calorie is usually defined as the amount of heat required to raise the temperature of one gram of water from 14.5 ° to 15.5°C. This unit is about the aver-age amount required to raise the temperature of one gram of water one degree at any point between 0° and 100 ° C. A more exact definition (generally accepted in the U.S.) is that one caloric equals 4.1840 absolute joules.
CAN'CEL , v. (1) To divide numbers (or factors) out of the numerator and denominator of a fraction;
the number 2 having been canceled out. (2) Two quantities of opposite sign but numerically equal are said to cancel when added; 2x + 3y-2x reduces to 2y, the terms 2x and — 2x having canceled out.
CAN'CEL-LA'TION, n. The act of dividing like factors out of numerator and denominator ofa fraction ; sometimes used of two quantities of different signs which cancel each other in addition. Also used for the process of eliminating z when replacing x + z = y + z by x = y, or xz = yz by x = y (if 
). See DOMAIN —integral domain, semi— semigroup.
CA-NON' I-CAL , adj. canonical random vari- ables. Given a set S of ? random variables and a set T of q random variables, there are sets of random variables 
and 
called canonical random variables, such that any two members of the same set have zero correlation, each 
has zero correlation with each 
for which 
, each 
is a linear combination of members of S, each 
is a linear combination of members of T 7 and each 
and each 
has mean 0 and variance 1. The correlations of 
with 
for 
(minimum of ? and q) are canonical correlations.
canonical form of a matrix. That which has been considered the simplest and most con venient form to which square matrices of a certain class can be reduced by a certain type of transformation. Syn. normal form. E.g. : (1) Any square matrix can be reduced by elementary operations or an equivalent trans- ormailon to the canonical form having non- zero elements only in the principal diagonal; or when the elements are polynomials (or integers, etc.) to Smith's canonical form having zeros except in the principal diagonal and each diagonal element being a factor of the next lower (if not zero). (2) Any matrix can be reduced by a collineatory transformation to the Jacobi canonical form having zeros below the principal diagonal and characteristic roots as elements of the principal diagonal, or to the classical canonical form having zeros except for a sequence of Jordan matrices situated along the principal diagonal. The exact type of the classical canonical matrix is specified by its Segre characteristic a set of integers which are the orders of the Jordan submatrices, those integers which correspond to submatrices containing the same charac- teristic root being bracketed together. When thecharacleristic roots are distinct, the classical canonical form is a diagonal matrix. (3) A symmetric matrix can be reduced to a diagonal matrix by a congruent transformation. ( |4) A normal matrix (and hence a Hermitian or a unitary matrix) can be reduced by a unitary transformation Io a diagonal matrix having characteristic roots along the principal diagonal.
canonical representation of a space curve in the neighborhood of a point. Representation of the curve in the neighborhood of the point P 0 , with the arc length from the point as pa rameter and the axes of the moving trihedral as coordinates axes. The representation has the form
where 
and 
are the radii of curvature and torsion, respeclively, at 
CAN'TI-LE'VER, adj. cantilever beam. A projecting beam supported at one end only.
CANTOR, Georg Ferdinand Ludwig Philipp (1845-1918). German set theorist (born in Russia, whither his father had migrated from Denmark before moving to Germany). In his day, his theories concerning infinite sets seemed revolutionary and created much controversy.
Cantor set . The set of numbers in the closed interval [0, 1 ] that are not in the open middle third of [0, 1] and not in the open middle third of any interval obtained by sep- arating [0, 1] into three equal intervals, then dividing each of these intervals into three equal intervals, etc. A number belongs to the Cantor set if and only if it has a repre sentation in the ternary system of type 
, where each d n is either 0 or 2. The Cantor set is perfect and nondense and all its points are frontier points. Also called the Cantor discontinuum , and the Cantor ternary set.
CAP , n . The symbol 
used to denote the intersection of two sets. See intersection.
CAP'I-TAL, adj., n. capital stock. The money invested by a corporation to carry on its business; wealth used in production, manufacturing, or business of any sort, which, having been so used, is available for use again. Capital stock may be disseminated by losses but is not consumed in the routine process of a business.
circulating capital . Capital consumed, or changed in form, in the process of production or of operating a business such as that used to purchase raw materials. Capital invested permanently such as that invested in build ings, machinery, etc., is fixed capital.
CAP'I-TAL-IZED , adj. capitalized cost. The sum of the first cost of an asset and the present value of replacements to be made perpetually at the ends of given periods.
CARATHEODORY , Constantin (1873- 19S0). German analyst who worked largely in complex-variable theory and the calculus of variations.
Caratheodory measure . See MEASURE.
Caratheodory's theorem . If S is a subset of n -dimensional space, then each point of the convex span of S is a convex combina tion of n + 1 (or fewer) points of S . See related theorems under Helly, RADON, STElNITZ.
CARDAN, Jerome (Girolamo Cardano) (1501-1576). Italian physician and mathe- matician.
Cardan's solution of the cubic . ? solu tion of the reduced cubic (see REDUCED — reduced cubic),
by the substitution 
will be a root of the equation if 
and 
or if 
is a root of the quadratic
equation in 
, 
and 
If 
is a cube root of 
and 
then the three roots of the reduced cubic are
where 
is a cube root of unity. This is equivalent to the formula 
where 
and the cube roots are to be chosen so that their product is 
The number R is negative if and only if the three roots of the cubic are real and distinct; this is called the irreducible case, since the formulas (although still correct) involve the cube roots of complex numbers. This general solution of the reduced cubic was completed by Tartaglia, who showed it Io Cardan. Cardan gave an oath of secrecy, but published the solution (giving credit to Tartaglia),
CAR'DI-NAL , adj. cardinal number . ? num ber which designates the manyness of a set of things; the number of units, but not the order in which they are arranged ; used in distinction to signed numbers. E.g., when one says 3 dolls, the 3 is a cardinal number. Tech. Two sets arc said to have the same cardinal number if their elements can be put inio one-to-one correspondence with each other. Thus a symbol or cardinal number can be associated with any set. The cardinal number of a set is also called the potency of the set and the power of the set {e.g., a set whose elements can be put into 1-1 correspondence with the real numbers is said to have the power of the continuum ). The cardinal number of the set 
is denoted by n. The cardinal number of all countably infinite sets is called Aleph-null or Aleph-zero and is designated by N 0 , and the cardinal number of all real num bers is designated by c. The cardinal number 2 C is the cardinal number of the set of all subsets of the real numbers (i.e., the set of all real-valued functions defined for all real numbers) and is greater than c in the sense that the real numbers can be put into one-to- one correspondence with a subset of the real functions but not conversely. See OR- DINAL-ordinal number, EQUIVALENT- equivalent sets.
CAR'DI-OID , n . The locus (in a plane) of a fixed point on a given circle which rolls on an equal but fixed circle. If a is the radius of the fixed circle, 
the vectorial angle, and r the radius vector where the pole is on the fixed circle and the polar axis is on a diameter of the fixed circle the polar equation of the cardioid is 
A cardioid is an epicycloid of one loop and a special case of the limacon.
CARTAN, Elie Joseph (1869-1951). French algebraist, group theorist, differential geom eter, and relativist. Worked on classification of Lie algebras and Lie groups, differential geometry in the large, and stability theory. Introduced exterior differential forms and spinors.
CARTAN, Henri Paul (1904- ). French analyst, algebraist, and topologist. Son of Elie Cartan. Worked particularly in algebraic topology, theory of analytic functions of one and several variables, theory of sheaves, and potential theory.
CAR-TE'SIAN , adj. Cartesian coordinates . In the plane , a point can be located by its distances from two intersecting straight lines, the distance from one line being measured along a parallel to the other line. The two intersecting lines are called axes {jr-axis and /-axis), oblique axes when they are not per pendicular, and rectangular axes when they are perpendicular. The coordinates are then
Rectangular Axis Oblique Axis called oblique coordinates and rectangular coordinates. respectively . The foordinale measured from the y-axis parallel to the x-axis is called the abscissa and the other coordinate is called the ordinate . In space , three planes (XOY, XOZ, and YOZ in the figure) can be used to locale points by giving their distance from each of the planes along a line parallel to the intersection of the other two. If the planes are mutually perpendicular, these distances are called the rectangular Cartesian coordinates of the point in space, or the rectangular or Cartesian coordinates . The lhrcc intersections of these three planes are called the axes of coordinates and arc usually labeled the x-axis, y-axis, and z-axis. Their common point is called the origin. The three axes are called a coordinate trihedral (sec tri- hedral). The coordinate planes separate space into eight compartments, called octants. The octant containing the three positive axes as edges is called the 1st octant (or coordinate trihedral ). The other octants are usually numbered 2, 3, 4, 5, 6, 7, 8; 2, 3, and 4 are reckoned counterclockwise around the positive z-axis (or clockwise if the coordinate system is left-handed), then the quadrant vertically beneath the first quadrant is labeled 5, and the remaining quadrants 6, 7, and 8, taken in counterclockwise (or clockwise) order as
before. A rectangular space coordinate is quite commonly thought of as the projection of the line from the origin to the point upon the axis perpendicular to the plane from which the coordinate is measured; i.e., x = OA, y = OB, and z = OC in the figure.
Cartesian product . See product — Car tesian product.
Cartesian space . Same as EUCLIDEAN SPACE.
CASH , n, Money of any kind; usually coin or paper money, but frequently includes checks, drafts, notes, and other sorts of commercial paper, which are immediately convertible into currency. cash equivalent of an annuity . Same as PRESENT VALUE. See SURRENDER.
CASSINI, Jean Dominique (1625-1712). French astronomer, geographer, and geometer.
Ovals of Cassini . The locus of the vertex of a triangle when the product of the sides adjacent to the vertex is a constant and the length of the opposite side is fixed. When the constant is equal to one-fourth of the square of the fixed side, the curve is called a lemniscale . If 
denotes the constant and a one-half the length of the fixed side, the Cartesian equation takes the form
If 
is less than 
the curve consists of two distinct ovals; if 
is greater than 
it con sists of one, and if 
is equal to 
it reduces to the lemniscate. The figure illustrates the case in which 
CAST'ING , n. c asting out nines . A method used to check multiplication (and sometimes division); based on the fact that the excess of nines in the product equals the excess in the product of the excesses in the multiplier and multiplicand. See excess. E.g., to check the multiplication 832x736 = 612,352, add the digits in 612,352, subtracting 9 as the sum reaches or exceeds 9. This gives 1. Do the same for 832, and for 736; the results are 4 and 7. Now multiply 4 by 7, getting 28. Then add 2 and. 8 and subtract 9. This leaves 1 which is the same excess that was obtained for the product. This method can also be used to check addition (or subtraction), since the excess of nines in a sum is equal to the excess in the sum of the excesses of the addends.
CAT'E-GO-RY , adj., n. (1) See below, category of sets. (2) A category K consists of two classes, 
and 
for which the members of 
are called objects , the members of 
are called morphisms , and the following conditions arc satisfied: (i) With each ordered pair (a, b) of objects there is associated a set 
of morphisms such that each member of 
belongs to exactly one of these sets; (ii) if/is in 
and g is in then 
the product or composite 
of f and g is defined uniquely and is a member of 
(iii) if f, g, and h are members of 
and 
respectively, so that 
and 
are defined, then 
for each object a, there is a morphism 
in 
called the identity mo rphism, such that 
and 
if there are objects b and c for which f is in 
and g is in 
The concept of category provides an abstract general model for many situations in which sets with certain structures are studied along with a class of mappings that preserve these structures. Examples of such categories are: (i) 
is the collection of all subsets of a set T and 
is the set of all functions which have domain a and whose ranges are con tained in the set b; (ii) 
is a collection of groups and 
is the set of all homomorphisms from the group a into the group b; (iii) 
is a collection of topological spaces and 
. is the set of all continuous functions which have domain a and whose ranges are contained in b . A zero in a category is an object 0 with the property that, for any object a , both 
and 
have exactly one member. If a category has a zero, then the zero morphism in 
is 
where 
and 
are the unique members of 
and 
respectively. An isomorphism or equivalence in 
is a morphism f in 
which has the property that there is a morphism g in 
such that 
and 
are the identity morphisms in 
and 
respectively. An isomorphism that belongs to 
is an automorphism and a morphism that belongs to 
is an endomorphism. See functor.
Baire's category theorem. The theorem which slates that a complete metric space is of second category in itself. An equivalent statement is that the intersection of any sequence of dense open sets in a complete metric space is dense. E.g., the space C of all functions which arc continuous on the closed interval [0, 1] is a complete metric space if the distance d(f, g) is defined to be the least upper bound of 
The set of all members of C which are different iable at one or more points of [0, 1] can be shown to be of first category in C, so that the set of con-tinuous functions not differentiable at any point of [0, 1] is of second category.
Banach's category theorem. The theorem which states that if a set S contained in a topological space 
is of second category in T , then there is a nonempty open set U in T such that S is of second category at every point of U. It follows From this theorem that a subset of T is of first category in T if it is of first category at each point of T.
category of sets. A set S is of first category in a set T if it can be represented as a countable union of sets each of which is nowhere dense in T. Any set which is not of the first category is said to be of second category . A set S is of first category at the point x if there is a neigh- borhood U of x such that the intersection of U and S is of first category. The complement of a set of first category in T is called a re- sidual set of T (sometimes residual set is used only for complements of sets of first category in sets T which have the property that every nonempty open set in T is of second category). A subset S of the real line is of the first category if and only if there is a one-to-one transformation of the line onto itself for which S corresponds to a set of measure zero which is also an 
set (see borel - Borel set).
CAT'E-NA-RY , n. The plane curve in which a uniform flexible cable hangs when sus pended from two points. Its equation in rectangular coordinates is 
where a is the y -intercept.
CAT'E-NOID , n. The surface of revolution generated by the rotation of a catenary about its axis. The only minimal surface of rev olution is the catenoid. See catenary,
CAUCHY, Augustin Louis (1789-1857). Great French analyst, applied mathema- tician, and group theorist. After EuIer, the most prolific mathematician in history. Worked in wave, elasticity, and group theory; introduced modern rigor into calcu- lus; founded theory of functions of a com- plex variable; inaugurated modern era in differential equations with his existence theorems.
Cauchy condensation test for conver- gence. If is a series of positive mono- tonic decreasing 
terms and ? is any positive integer, then the series 
and
are either both convergent or both divergent.
Cauchy condition for convergence of a sequence. An infinite sequence converges if, and only if, the numerical difference between every two of its terms is as small as desired, provided both terms arc sufficiently Tar out in the sequence. Tech. The infinite sequence 
converges if, and only if, for every 
there exists an N such that 
for all 
and all 
See sequence — Cauchy sequence. Same as cauchy con dition for CONVERGENCE of a series when 
is looked upon as the sum to n terms of the series 
Cauchy condition for convergence of a series. The sum of any number of terms can be made as small as desired by starting sufficiently far out in the series. Tech. A necessary and sufficient condition for convergence of a series 
is that, for any 
there exists an N such that 
for all 
and all 
See complete — complete space.
Cauchy distribution . A random variable has a Cauchy distribution or is a Cauchy random variable if there are numbers u and L for which the probability density func tion f satisfies 
The distribution is symmetric about u, but it has no mean or variance since it has no finite moments of any order. The dis- tribution function is 
arc tan 
The means of random samples of order n of a Cauchy random var- iable X have the same distribution as X for all n . The t distribution with one degree of freedom is a Cauchy distribution with L. = 1 and u = 0.
Cauchy-Hadamard theorem. The theorem states that the radius of convergence of the Taylor scries 
in the complex variable z is given by
Cauchy-Riemann partial differential equa- tions. For functions u and ? of x and y, the ( Cauchy-Reimann equations are 
and 
The equations characterize analytic functions 
of thecomplex variable 
and are satisfied if and only if the map T defined by 
is conforma/ except at points where all four partial derivatives vanish.
Cauchy sequence . See sequence — Cauchy sequence.
Cauchy's form of the remainder for Taylor's theorem. See taylor —Taylor's theorem.
Cauchy's inequality. The inequality
Also see SCHWARZ. -Schwarz's inequality. Cauchy's integral formula. The formula
where f is an analytic function of the complex variable z in a finite simply connected domain D, C is a simple cloned rectifiable curve in D , and z is a point in the finite domain bounded by C . This formula can be extended to the following, for n any positive integer:
Cauchy's iniegral test for convergence of an infinite series. Suppose that for the series 
there is a function f which has the proper- ties: (i) there is a. number N such that f is a monotonically decreasing positive function on the interval consisting of all numbers greater than N, and (ii) 
for all n sufficiently large. Then a necessary and sufficient condi tion for convergence of the series, 
is that there exists a number a such that 
converges, in the case of the ? series,
* 1,
and 
Hence the ? series converges for 
anddiverges for 
Cauchy's integral theorem. If f is analytic in a finite simply connected domain D of the complex plane, and C is a closed rcctifiable curve in D, then
Cauchy's mean-value formula . See MEAN —second mean-values theorem.
Cauchy's ratio and root tests . See RATIO-ratio test, ROOT-root test.
CAVALIERI, Francesco Bonaventura ( 1598-1647). Italian physicist and mathematician. Further developed Archimedes' method of exhaustion, thus anticipating the invention of the integral calculus.
Cavalieri's theorem. If two solids have equal altitudes and all plane sections paral- lel to their bases and at equal distances from their bases have equal areas, then the solids have the same volume.
CAYLEY. Arthur (1821-1895). English algebraist, geometer and analyst. Contrib- uted especially to the theory of algebraic in- variants and higher-dimensional geometry.
See SYLVESTER.
Cayley algebra. The set of symbols of type A + Be, where A and B are quaternions and addition and multiplication are defined by
with 
and 
the conjugates of the quaternions C and D. Except for multiplication not being associative, the Cayley algebra satisfies all axioms for a division algebra with unit element. As a vector space over the field of real numbers, it is of dimension 8 with the basis 
The definition of multiplication implies that 
but that 
and 
The members of the Cayley algebra are Cayley numbers . Sec frobenius Frobenius' theorem,
Cayley's theorem . Any group is isomorphic to a group of transformations. In particular, a group G is isomorphic to a permutation group on the set G .
CECH, Eduard (1893-1960). Polish topolo- gist and projective differential geometer. Stone-Cech compactification . See COMPACTIFIC ATION.
CE-LES'TIAL , adj. Of or pertaining to the skies or heavens.
altitude of a celestial point. See altitude.
celestial equator. See HOUR -hour angle and hour circle, equator.
celestial horizon, meridian, and pole. See hour hour angle and hour circle.
celestial sphere. The conceptual sphere on which all the celestial objects are seen in pro jection and appear to move.
CELL , n. An n -dimensional cell ( n -cell) is a set which is homeomorphic either with the set of points 
of n -dimensional
Euclidean space for which 
or with the set for which 
(it is an open
n -cell in the first case, a closed n -cell in the other). A 0-cell is a point; a 1-cell is an open or a closed interval or a continuous defor- mation of an open or a closed interval. Circles (or simply polygons) and their interiors are examples of closed 2-cells; spheres (or simple polyhedrons) and their interiors are closed 3-cells. A closed n -cell is sometimes called a solid n-sphere, or an n-disk.
CEN'TER, n. Usually the center of symmetry, such as the center of a circle, or the center of a regular polygon as the center of the inscribed circle. See CIRCLE, ellipse, ellipsoid, HYI'erboloid, symmetric- symmetric geometric configurations. center of attraction . Same as center OF MASS.
center of curvature. See CURVATURE— curvature of a curve , curvature of a surface. geodesic geodesic curvature.
center of a curve . The point (if it exists) about which the curve is symmetrical. Curves such as the hyperbola, which are not closed, but are symmetrical about a given point, are said to have this point as a center, but the term center commonly refers to closed curves such as circles and ellipses. Syn. center of symmetry. See symmetry — symmetry of a geometric configuration.
center of gravity . Same as center of mass.
center of mass . The point at which a mass (body) can be considered as being concen trated without altering the effect of the attraction that the earth has upon it; the point in a body through which the resultant of the gravitational forces, acting on all its particles, passes regardless of the orientation of the body; the point about which the body is in equilibrium; the point such that the moment about any line is the same as it would be if the body were concentrated at that point. See MOMENT — moment of mass. The center of mass is the point of the body which has the same motion that a particle having the mass of the whole body would have if the resultant of all the forces acting on the body were applied to it. If the body consists of a set of particles, the center of mass is the point determined by
the vector 
where 
is the position vector of the mass 
in the system of particles 
In case of a continuous distribution of mass, the vector 
locating the center of mass of a body is given by
where the integration is carried out throughout the space 
occupied by the body. The coordinates 
and 
of the center of mass are given by
where m is the total mass of the body, x, y, ? are the coordinates of some point in the element of mass, 
and 
indicates that the integration is to be taken over the entire body, the integration being single, double, or triple depending upon the form of 
may, for instance, be one of the forms: p ds, px dy, 
where 
is density. If elements such as 
or . 
are used, one must take for the point in the element of mass the approximate center of mass of these strips (elements). Syn. center of attraction, center of gravity. See centroid.
center of pressure of a surface submerged in a liquid. That point at which all the force could be applied and produce the same effect as when the force is distributed.
center of a sheaf. See SHEAF — sheaf of planes.
center of similarity (or similitude) of two configurations. See radially- radially re- lated, figures.
radical center . See radical.
CEN - TES'I-MAL , adj. centesimal system of measuring angles. The system in which the right angle is divided into 100 equal parts, called degrees, a degree into 100 minutes and a minute into 100 seconds. Not in common use.
CEN'TI-GRADE , adj. centigrade thermom eter. A thermometer on which 
and 
respectively, indicate the freezing and boiling points of water. Formulas for converting from a temperature 
measured on the centi grade scale to the corresponding temperature 
as measured on the Fahrenheit scale, and conversely, are 
CEN'TI-GRAM , n. One hundredth of a gram. See denominate NUMBERS in the appendix.
CEN'TI-ME'TER , n. One hundredth part of a meter. See denominate numbers in the appendix.
CEN'TRA L, adj., n. central angle in a circle . An angle whose sides are radii. An angle with its vertex at the center. See figure under CIRCLE.
central conics . Ellipses and hyperbolas.
central death rate . See death.
central of a group. The set of all elements of the group which commute with every e lement of the group. The central is an invariant subgroup, but may be contained properly in an invariant subgroup. See GROUP.
central limit theorem (Statistics). Given a sequence 
of independent random variables, a central limit theorem is anv theorem which gives conditions for to have approximately
a 
normal distribution when n is large, where 
and 
are the mean and variance of For example, if all the random var iables 
nave the same distribution function with finite mean 
and finite variance 
then the distribution of 
approaches uniformly 
the normal distribution with mean and variance 1 as 
If in particular each 
is a Ber noulli experiment with probability ? of suc cess, then 
ap proaches uniformly the normal distribution with mean and variance 1, where s(n) is the number of successes in ? experiments (see binomial -binomial distribution).
central moment . See MOMENT-moment of a distribution.
central plane and point of a ruling on a ruled surface. See ruling.
central projection. See projection.
central quadries . Quadrics having centers — ellipsoids and hypcrboloids.
measures of central tendency . Sec MEASURE — measures of central tendency.
CEN-TRIF'U-GAL , adj. centrifugal force .
1) The force which a mass, constrained to move in a path, exerts on the constraint in a direction along the radius of curvature.
2) A particle of mass m, rotating with the angular velocity 
about a point 
at a distance 
from the particle, is subjected to a force, called centrifugal force , of magnitude 
where 
is the speed of the particle relative to 
). The direction of this force on the particle is awey from the center of rotation. The equal and oppositely directed force is tailed centripetal force .