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Roberts's list of southern variables (ibid.

164Roberts's list of southern variables (ibid.

164Pierpont, Lectures on the Theory of Functions of Real Variables (1905).

95Resultants.-When we are given k homogeneous equations in k variables or k non-homogeneous equations in k - i variables, the equations being independent, it is always possible to derive from them a single equation R = o, where in R the variables do not appear.

67Symbolic Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az wherein al, a2 are umbrae, such that n-1 n-1 n a 1, a 1 a 2, ...a 1 a 2, a2 are symbolical respreentations of the real coefficients ï¿½o, ai,...

51This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression -9m 6 (x 3 +y 3 +z 3) 2 - (2m +5m 4 +20m 7) (x3 +y3+z3)xyz - (15m 2 +78m 5 -12m 8) Passing on to the ternary quartic we find that the number of ground forms is apparently very great.

43If the variables of the quantic f(x i, x 2) be subjected to the linear transformation x1 = a12Et2, x2 = a21E1+a2252, E1, being new variables replacing x1, x 2 and the coefficients an, all, a 21, a22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic f% 1tn n n-1 n-2 52) = a S +(1)a11 E 2 + (2)a2E1 E 2 +ï¿½ï¿½ï¿½ in the new variables which is of the same order as the original quantic; the new coefficients a, a, a'...a are linear functions 0 1 2 n of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.

44The machine will figure this out as it collects more data and incorporates more variables, and then experiments on people to see which combinations of factors work the best.

44Turner has analysed harmonically the light-curves of a number of long-period variables, and has shown that when they are arranged in a natural series the sun takes its place in the series near, but not actually at, one end.

45Thus we shall be able to specify the system completely when the number of variables, viz.

31If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.

33A particular quantic of the system may be of the same or different degrees in the pairs of variables which it involves, and these degrees may vary from quantic to quantic of the system.

33- An important method for the formation of covariants is connected with the form f +X4), where f and 4 are of the same order in the variables and X is an arbitrary constant.

22- zn +9 1 -z2.1 -z3....1-z8; and since this expression is unaltered by the interchange of n and B we prove Hermite's Law of Reciprocity, which states that the asyzygetic forms of degree 0 for the /t ie are equinumerous with those of degree n for the The degree of the covariant in the variables is e=nO-2w; consequently we are only concerned with positive terms in the developments and (w, 0, n) - (w - r; 0, n) will be negative unless nO It is convenient to enumerate the seminvariants of degree 0 and order e=n0-2w by a generating function; so, in the first written generating function for seminvariants, write z2 for z and az n for a;.

22Instead of following the motion of each individual part of a material system, he showed that, if we determine its configuration by a sufficient number of variables, whose number is that of the degrees of freedom to move (there being as many equations as the system has degrees of freedom), the kinetic and potential energies of the system can be expressed in terms of these, and the differential equations of motion thence deduced by simple differentiation.

22We cannot combine the equations so as to eliminate the variables unless on the supposition that the equations are simultaneous, i.e.

23It is almost impossible to execute a pure controlled study of anything relating to nutrition because there are simply too many variables to consider.

25Computers, especially computers of the future, will have no trouble handling all the variables that influence nutrition, though there will be millions of them.

25The process of transvection is connected with the operations 12; for?k (a m b n) = (ab)kam-kbn-k, (x y x y or S 2 k (a x by) x = 4))k; so also is the polar process, for since f k m-k k k n - k k y = a x by, 4)y = bx by, if we take the k th transvectant of f i x; over 4 k, regarding y,, y 2 as the variables, (f k, 4)y) k (ab) ka x -kb k (f, 15)k; or the k th transvectant of the k th polars, in regard to y, is equal to the kth transvectant of the forms. Moreover, the kth transvectant (ab) k a m-k b: -k is derivable from the kth polar of ax, viz.

29The process of transvection is connected with the operations 12; for?k (a m b n) = (ab)kam-kbn-k, (x y x y or S 2 k (a x by) x = 4))k; so also is the polar process, for since f k m-k k k n - k k y = a x by, 4)y = bx by, if we take the k th transvectant of f i x; over 4 k, regarding y,, y 2 as the variables, (f k, 4)y) k (ab) ka x -kb k (f, 15)k; or the k th transvectant of the k th polars, in regard to y, is equal to the kth transvectant of the forms. Moreover, the kth transvectant (ab) k a m-k b: -k is derivable from the kth polar of ax, viz.

29Let a covariant of degree e in the variables, and of degree 8 in the coefficients (the weight of the leading coefficient being w and n8-2w = ï¿½), be Coxl -}- ec l l 1 x 2 -{-...

10Suppose n dependent variables yl, y2,ï¿½ï¿½ï¿½yn, each of which is a function of n independent variables x1, x2 i ï¿½ï¿½ï¿½xn, so that y s = f s (x i, x 2, ...x n).

11Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.

11Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables.

11CY The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables.

11Discriminants.-The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.

11in addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.

11If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...

11v., established the important result that in the case of a form in n variables, the concomitants of the form, or of a system of such forms, involve in the aggregate n-1 classes of aa =5135 4 +4B8 3 p) =0, =5(135 4 - 4A 2 p 4) =0, P yield by elimination of S and p the discriminant D =64B-A2.

11Variables of the Algol class are rendered difficult to discover by the incidental character of their fluctuations.

11corum, a double star, of magnitudes 3 and 6; this star was named Cor Caroli, or The Heart of Charles II., by Edmund Halley, on the suggestion of Sir Charles Scarborough (1616-1694), the court physician; a cluster of stars of the firth magnitude and fainter, extremely rich in variables, of the goo stars examined no less than 132 being regularly variable.

11To these must be added the external variables of temperature and pressure, and then as the total number of variables, we have r (n+I) + 2.

11There are several wellmarked varieties of short-period variables; the most important are typified by the stars Algol, # Lyrae, Geminorum and S Cephei.

11There are several wellmarked varieties of short-period variables; the most important are typified by the stars Algol, # Lyrae, Geminorum and S Cephei.

11Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.

12For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.

12The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.

12If, however, F involve as well the variables, viz.

12are contragredient with the d- variables x, y, z, ...

12of the symbolic factors of the form are replaced by IA others in which new variables y1, y2 replace the old variables x1, x 2 .

12Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.

12Stars having this type of spectrum are always variable, and a large proportion of the more recently discovered long-period variables have been detected through their characteristic spectrum.

12Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.

12Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;...

14To determine these variables we may form equations between the chemical potentials of the different components - quantities which are functions of the variables to be determined.

17Since the two expressions (9) are the partial differential-coefficients of a single function E of the independent variables v and 0, we shall obtain the same result, namely d 2 E/d0dv, if we differentiate the first with respect to v and the second with respect to 0.

19Taking the variables to be x, y and effecting the linear transformation x = X1X+1.11Y, y = X2X+It2Y, X 2 +Y2X Y Xl - X2 y = _ x X I + AI R X 122 so that - ï¿½l b it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence.

110A similar expression for the variation of the specific heat S at constant pressure is obtained from the second expression in (8), by taking p and 0 as independent variables; but it follows more directly from a consideration of the variation of the function (E+pv).

110We thus recognize two classes of variables, of which (I) the long-period variables have periods ranging in general from 150 to 450 days, though a few are outside these limits, and (2) the short-period variables have periods less than 50 days (in the majority of cases less than io days).

111This system is of much service in following out mathematical, physical and chemical problems in which it is necessary to represent four variables.

00It might be thought that the "futures" of different months, being substitutes in proportion to their temporal proximity to one another, should vary together exactly; but it would seem to be a sufficient reply that as they are not perfect substitutes they are in some slight degree independent variables.

00Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00ALGEBRAIC FORMS. The subject-matter of algebraic forms is to a large extent connected with the linear transformation of algebraical polynomials which involve two or more variables.

00each of them satisfied by a common system of values; hence the equation R =o is derived on this supposition, and the vanishing of R expresses the condition that the equations can be satisfied by a common system of values assigned to the variables.

00Schlafli 1 this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables.

00THE Theory Of Binary Forms A binary form of order n is a homogeneous polynomial of the nth degree in two variables.

00If the form, sometimes termed a quantic, be equated to zero the n+I coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.

00Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables.

00There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.

00All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.

00k k According to the well-known law for the changes of independent variables.

00The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation of the variables.

00The idea_can be generalized so as to have regard to ternary and higher forms each of the same order and of the same number of variables.

00variables.

00Q 1 The Unreduced Generating Function Which Enumerates The Covariants Of Degrees 0, 0' In The Coefficients And Order E In The Variables.

00For further details of his mathematical investigations see the articles Theory of groups, and Functions Of Complex Variables.

00BESSEL FUNCTION, a certain mathematical relation between two variables.

00The number F is called the number of degrees of freedom of the system, and is measured by the excess of the number of unknowns over the number of variables.

00This was attempted variables.

00Besides the shortness of the period these variables possess other characteristics which differentiate them from the long-period variables.

00In the Algol variables one of the component stars is dark (that is to say, dark in comparison with the other), and once in each revolution, passing between us and the bright component, partially hides it.

00This class of variables is accordingly characterized by the fact that for the greater part of the period the star shines steadily with its maximum brilliancy, but fades away for a short time during each period.

00About 56 Algol variables were known in 1907; the variables of this class are the most difficult to detect, for the short period of obscuration may easily escape notice unless the star is watched continuously.

00When the two stars are of equal brilliancy the minima are equal; this is the case in variables of the Geminorum type.

00A large eccentricity also produces an unsymmetrical light variation, the minimum occurring at a time not midway between two maxima; stars of this character are called Cepheid variables, after the typical star S Cephei.

00All the best-known short-period variables have been proved to be binary systems spectroscopically, and to have periods corresponding with the period of light variation, so that to this extent the hypothesis we have described is well founded; but it is doubtful if it is the whole explanation.

00No hard and fast physical distinction can be drawn between the various classes of short-period variables; as the distance between the components diminishes the Algol variable merges insensibly into the (3 Lyrae type.

00Certain clusters contain a remarkable number of short-period variables.

00Thus the cluster Messier 5 was found at Harvard to contain 185 variables out of 900 stars examined.

00Moreover, the light-curves were all of a uniform type, a distinctive feature of " cluster variables " being the rapid rise to a maximum and slow decline.

00Stars of the class to which the Algol type of variables belongs will appear to us to vary only in the exceptional case when the plane of the orbit passes so near our sun that one body appears to pass over the other and so causes an eclipse.

00A continuous gradation can be traced from the most widely separated visual binaries, whose periods are many thousand years, to spectroscopic binaries, Algol and # Lyrae variables, whose periods are a few hours and whose components may even be in contact, and from these to dumb-bell shaped stars and finally to ordinary single stars.

00or "Antarian " stars are of a reddish colour, such as Antares, Betelgeux, Mira, and many of the long-period variables.

00The phenomena of long-period variables show that the surface brilliancy may vary very greatly, even in the same star.

00For the Algol variables it is possible to form even more direct calculations of the density, for from the duration of the eclipse an approximate estimate of the size of the star may be made.

00Roberts concluded in this way that the average density of the Algol variables and their eclipsing companions is about one-eighth that of the sun.

004, together constitute a catalogue of 3734 variable stars; ephemerides of over 800 variables are given in the Vierteljahrsschrift of the Astronomische Gesellschaft.

00The series of equations of the type (3) is then replaced by a single linear partial differential equation, or by a set of two or three such equations, according to the number of dependent variables.

00These variables represent the whole assemblage of generalized co-ordinates qr; they are continuous functions of the independent variables x, y, 1 whose range of variation corresponds to that of the index r, and of 1.

00For example, in a one-dimensional system such as a string or a bar, we have one dependent variable, and two independent variables x and t.

00To determine the free oscillations we assume a time factor e~1 the equations then become linear differential equations between the dependent variables of the problem and the independent variables x, or x, y, or x, y, 1 as the case may be.

00If the range of the independent variable or variables is unlimited, the value of in is at our disposal, and the solution gives us the laws of wave-propagation (see WAVE).

00and the corresponding axes parallel, then by changing the signs of x x, y, the values x',17; x', y' must likewise change their sign, but retain their arithmetical values; this means that the series are restricted to odd powers of the unmarked variables.

00Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).

00This concept is extended to algebra: since a line, surface and solid are represented by linear, quadratic and cubic equations, and are of one, two and three dimensions; a biquadratic equation has its highest terms of four dimensions, and, in general, an equation in any number of variables which has the greatest sum of the indices of any term equal to n is said to have n dimensions.

00r 1.2 r1.3 r2.3 The theorems of motion just cited are expressed by seven integrals, or equations expressing a law that certain functions of the variables and of the time remain constant.

00We have supposed this to be done at a certain point P of the orbit, the direction and speed being expressed by the variables x, y, x' and y'.

00Now, consider the values of these same variables expressing the position of the planet at a second point Q, and the speed with which it passes that point.

00Since the orbit is unchanged so long as no disturbing force acts, it follows that the elements determined by means of the two sets of values of the variables are in this case the same.

00In a word, although the position and speed of the planet and the direction of its motion are constantly changing, the values of the elements determined from these variables remain constant.

00This fact is fully expressed by the equations (4) where we have constants on one side of the equation equal to functions of the variables on the other.

00Functions of the variables possessing this property of remaining constant are termed integrals.

00The reason for taking the elements as the variables is that they vary very slowly, a property which facilitates their determination, since the variations may be treated as small quantities, of which the squares and products may be neglected in a first solution.

00When the varying elements are known these are computed by the equations (2) because, from the nature of the algebraic relations, the slowly varying elements are continuously determined by the equations (4), which express the same relations between the elements and the variables as do the equations (2) and (3).

00In all that precedes we have considered only two variables as determining the position of the planet, the latter being supposed to move in a plane.

00In mathematics the term has received special meanings; in mathematical tables the "argument" is the quantity upon which the other quantities in the table are made to depend; in the theory of complex variables, e.g.

00When light from an extended source is made to converge upon the crystal, the phenomenon of rings and brushes localized at infinity is obtained.The exact calculation of the intensity in this case is very complicated and the resulting expression is too unwieldy to be of any use, but as an approximation the formula for the case of a parallel beam may be employed, the quantities and p therein occurring being regarded as functions of the angle and plane of incidence and consequently as variables.

00The pronouns function semantically as variables bound by their quantifier antecedents.

00A further release of data, which contains additional derived variables, will be made available at a later date.

00The counterexample is the quantum exterior algebra in 2 variables.

00The higher the number of variables the longer the optimization algorithms will take to run.

00Nigel Hall Manchester Metropolitan University Variables in child and adolescent language brokering.

00Three TMD related symptoms and reported bruxism were used as dependent variables.

00See also categorical predictor variables, design matrix; or General Linear Models.

00only categorical (nominal) variables or variables with a relatively small number of different meaningful values should be crosstabulated.

00collocation scheme using uniform norms that compare values of field variables at two successive mesh iterations.

00Sets of variables engage in health insurance provider colorado some.

00Discrete Random Variables: mean and variance; linear combinations; covariance and correlation.

00confounding variables may be influencing results.

00In addition constrained optimization is covered (the variables may be subject to bounds or equality constrained optimization is covered (the variables may be subject to bounds or equality constraints, etc.

00constructing the bridge would have temporarily changed certain stream variables.

00correlated variables?

00This paper proposes a new test of independence based on the maximum canonical correlation between pairs of discrete variables.

00correlations between the variables are presented in matrix form in Table 2.

00Mantel statistics may turn out significant even if they have values that would seem very low for a Pearson product-moment correlation between two variables.

00Main outcome measures: Relationship between variables examined by calculating Pearson's product moment correlation coefficients.

00coupling between the two coils is represented by the factor k. There are clearly several variables in this arrangement.

00covariances of exogenous variables.

00explicit declaration of all variables is considered good modern programming practice.

00defineroceed as is usual for modeling a problem, by defining variables.

00The main dependent variables were levels of fear and worry in prisoners and officers.

00This in turn will simplify the derivation of design principles based on these variables.

00differential equations of high order, using integral measurements as its variables, does much the same thing.

00You can examine whether or not variables are normally distributed with histograms of frequency distributions.

00The good news is that Perl doesn't always use a dollar sign to identify variables.

00Donald script in eden, it is necessary to associate display actions with variables.

00See ENVIRONMENT VARIABLES for detailed descriptions of the mailx variables. sh ell [shell-command] !

00explanatory variables are included in the models.

00extern variables are assigned values in the master file when configuring drivers or modules.

00extraneous variables for the findings of either to be applied rigidly across the board.

00These variables are now used to dynamically generate the SiteMap.

00hemostatic variables increased with age and smoking habit.

00hypothesized processes or variables (theoretical constructs) poses a challenge by itself.

00independent variables used in the univariable analysis are listed in the appendix of the paper.

00The two main independent variables were the prison wing and the category of offense.

00instance variableS An anonymous array or anonymous hash can be used to hold instance variables.

00The expression may contain variables of the form " { i } " where i is a nonnegative integer.

00isolines of these two variables are shown in black.

00linear equationribed by linear dynamic equations subject to linear inequalities involving real and integer variables.

00Thus, language planning is a very complex process, involving linguistic, social, psychological, environmental and other variables.

00manipulated variables which may be used in control loops.

00M-1 dummy variables.

00multiple regression, the independent variables may be correlated.

00People's welfare depends upon real variables, not nominal variables.

00It is assumed that the data (for the variables) represent a sample from a multivariate normal distribution.

00Create variables, use variables in assignment statements and write code to carry out simple arithmetic operations.

00overlaps with other variables, pointees and references, a program may run incorrectly.

00Group differences in average volume of ischaemic penumbra salvaged will be analyzed by t-test, adjusting for baseline differences in prognostic variables.

00pertinent regulatory rules, backup volume, time requirements and other variables.

00The range for x can be further extended by using quadruple precision for the input x and related variables (see LONG WRITE-UP ).

00prognostic variables.

00quantifier rules provide no indication as to what terms or free variables must be used in their deployment.

00rationale for the selection of variables to be measured was (unfortunately) implicit rather than explicit.

00read-only variables, command will have to restore the old value of the variable.

00recourse variables would represent the liquidation (selling) of assets to meet liabilities.

00Given two new first-order variables a and b, the translations ST a and ST b are defined by mutual recursion.

00relationships between continuous variables were examined with Pearson product moment correlation coefficients.

00roots of a polynomial of degree d in n variables.

00socioeconomic variables the excess risks were lowered.

00static variables can be used to emulate constants, values that don't change.

00subscript expression occurring in the left part variables are evaluated in sequence from left to right.

00symptomatology variables were differentially associated with the four factors for both males and females.

00tweeter level control accommodates a range of room acoustic variables.

00unbind any variables exactly at the places in the tree where they were bound.

00unbound variables are allowed. names represent signals in syndi and are syntactically similar to identifiers.

00universalize the process and undervalue these important variables?

00unset variables.

00Bespoke languages are deemed necessary in order to avoid introducing confounding variables.

00You can declare variables at the start of any compound statement.

00We proceed as is usual for modeling a problem, by defining variables.

00A further release of data, which contains additional derived variables, will be made available at a later date.

00The number of manipulated variables which may be used in control loops.

00static variables can be used to emulate constants, values that don't change.

00These are all models that have categorical variables, as it's major components.

00Starting with the two dummy variables less likely to these are the.

00These environment variables may be used to control the executable's behavior.

00The integer variables LINES and COLS are defined in curses.h and will be filled in by initscr with the size of the screen.

00Keywords OBJ Set this keyword to free object heap variables only.

00instance variableS An anonymous array or anonymous hash can be used to hold instance variables.

00This system is of much service in following out mathematical, physical and chemical problems in which it is necessary to represent four variables.

00Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.

00It might be thought that the "futures" of different months, being substitutes in proportion to their temporal proximity to one another, should vary together exactly; but it would seem to be a sufficient reply that as they are not perfect substitutes they are in some slight degree independent variables.

00Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.

00For the subjects of this heading see the articles DIFFERENTIAL EQUATIONS; FOURIER'S SERIES; CONTINUED FRACTIONS; FUNCTION; FUNCTION OF REAL VARIABLES; FUNCTION COMPLEX; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; MAXIMA AND MINIMA; SERIES; SPHERICAL HARMONICS; TRIGONOMETRY; VARIATIONS, CALCULUS OF.

00ALGEBRAIC FORMS. The subject-matter of algebraic forms is to a large extent connected with the linear transformation of algebraical polynomials which involve two or more variables.

00When a homogeneous polynomial is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables.

00The investigation of the properties of these functions, as well for a single form as for a simultaneous set of forms, and as well for one as for many series of variables, is included in the theory of invariants.

00Suppose n dependent variables yl, y2,Ã¯¿½Ã¯¿½Ã¯¿½yn, each of which is a function of n independent variables x1, x2 i Ã¯¿½Ã¯¿½Ã¯¿½xn, so that y s = f s (x i, x 2, ...x n).

00Ã¯¿½ Oxl d 2x 77n If we have new variables z such that zs=4s(yl, Y2,...yn), we have also z s =1 Y 8(x1, x2,Ã¯¿½Ã¯¿½Ã¯¿½xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .

00Resultants.-When we are given k homogeneous equations in k variables or k non-homogeneous equations in k - i variables, the equations being independent, it is always possible to derive from them a single equation R = o, where in R the variables do not appear.

00We cannot combine the equations so as to eliminate the variables unless on the supposition that the equations are simultaneous, i.e.

00each of them satisfied by a common system of values; hence the equation R =o is derived on this supposition, and the vanishing of R expresses the condition that the equations can be satisfied by a common system of values assigned to the variables.

00Resultant Expressible as a Determinant.-From the theory of linear equations it can be gathered that the condition that p linear equations in p variables (homogeneous and independent) may be simultaneously satisfied is expressible as a determinant, viz.

00Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables.

00CY The proof being of general application we may state that a system of values which causes the vanishing of k polynomials in k variables causes also the vanishing of the Jacobian, and in particular, when the forms are of the same degree, the vanishing also of the differential coefficients of the Jacobian in regard to each of the variables.

00The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.

00If three equations, each of the second degree, in three variables be given, we have merely to eliminate the six products x, 2, z 2, yz, zx, xy from the six equations u = v = w = o = oy = = 0; if we apply the same process :to thesedz equations each of degree three, we obtain similarly a determinant of order 21, but thereafter the process fails.

00Discriminants.-The discriminant of a homogeneous polynomial in k variables is the resultant of the k polynomials formed by differentiations in regard to each of the variables.

00Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...

00Schlafli 1 this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables.

00THE Theory Of Binary Forms A binary form of order n is a homogeneous polynomial of the nth degree in two variables.

00If the form, sometimes termed a quantic, be equated to zero the n+I coefficients are equivalent to but n, since one can be made unity by division and the equation is to be regarded as one for the determination of the ratio of the variables.

00If the variables of the quantic f(x i, x 2) be subjected to the linear transformation x1 = a12Et2, x2 = a21E1+a2252, E1, being new variables replacing x1, x 2 and the coefficients an, all, a 21, a22, termed the coefficients of substitution (or of transformation), being constants, we arrive at a transformed quantic f% 1tn n n-1 n-2 52) = a S +(1)a11 E 2 + (2)a2E1 E 2 +Ã¯¿½Ã¯¿½Ã¯¿½ in the new variables which is of the same order as the original quantic; the new coefficients a, a, a'...a are linear functions 0 1 2 n of the original coefficients, and also linear functions of products, of the coefficients of substitution, of the nth degree.

00In the theory of forms we seek functions of the coefficients and variables of the original quantic which, save as to a power of the modulus of transformation, are equal to the like functions of the coefficients and variables of the transformed quantic. We may have such a function which does not involve the variables, viz.

00If, however, F involve as well the variables, viz.

00Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables.

00which have different coefficients, the same variables, and are of the same or different degrees in the variables; we may transform them all by the same substitution, so that they become _, _, _, _, _, _, f(a Ã‚°, a, a 2, ...; (b 0, b, b 2, ...; 1, S2),....

00Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;...

00in addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.

00A particular quantic of the system may be of the same or different degrees in the pairs of variables which it involves, and these degrees may vary from quantic to quantic of the system.

00Symbolic Form.-Restricting consideration, for the present, to binary forms in a single pair of variables, we must introduce the symbolic form of Aronhold, Clebsch and Gordan; they write the form Iln n n-1 n-1 n n n aixi+a2x2) = 44+(1) a l a 2 x 1 x2+...+a2.x2=az wherein al, a2 are umbrae, such that n-1 n-1 n a 1, a 1 a 2, ...a 1 a 2, a2 are symbolical respreentations of the real coefficients Ã¯¿½o, ai,...

00are contragredient with the d- variables x, y, z, ...

00If u, a quantic in x, y, z, ..., be expressed in terms of new variables X, Y, Z ...; and if, n,, ..., be quantities contragredient to x, y, z, ...; there are found to exist functions of, n, ?, ..., and of the coefficients in u, which need, at most, be multiplied by powers of the modulus to be made equal to the same functions of E, H, Z, ...

00There also exist functions, which involve both sets of variables as well as the coefficients of u, possessing a like property; such have been termed mixed concomitants, and they, like contravariants, may appertain as well to a system of forms as to a single form.

00that occur in the symbolic expression; the degree in the variables (i.e.

00of the symbolic factors of the form are replaced by IA others in which new variables y1, y2 replace the old variables x1, x 2 .

00The operation of taking the polar results in a symbolic product, and the repetition of the process in regard to new cogredient sets of variables results in symbolic forms. It is therefore an invariant process.

00All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.

00Every symbolic product, involving several sets of cogredient variables, can be exhibited as a sum of terms, each of which is a polar multiplied by a product of powers of the determinant factors (xy), (xz), (yz),...

00It is (f = (ab) 2 a n-2 r7 2 =Hx - =H; unsymbolically bolically it is a numerical multiple of the determinant a2 f a2f (32 f) 2Ã¯¿½ It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz.

00In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " Ã¯¿½Ã¯¿½ ' axn and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in I, 2, 3 variables respectively.

00- An important method for the formation of covariants is connected with the form f +X4), where f and 4 are of the same order in the variables and X is an arbitrary constant.

00k k According to the well-known law for the changes of independent variables.

00The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation of the variables.

00A leading proposition states that, if an invariant of Xax and i ubi be considered as a form in the variables X and, u, and an invariant of the latter be taken, the result will be a combinant of cif and b1'.

00The idea_can be generalized so as to have regard to ternary and higher forms each of the same order and of the same number of variables.

00the quartic to a quadratic. The new variables y1= 0 are the linear factors of 0.

00This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression -9m 6 (x 3 +y 3 +z 3) 2 - (2m +5m 4 +20m 7) (x3 +y3+z3)xyz - (15m 2 +78m 5 -12m 8) Passing on to the ternary quartic we find that the number of ground forms is apparently very great.

00v., established the important result that in the case of a form in n variables, the concomitants of the form, or of a system of such forms, involve in the aggregate n-1 classes of aa =5135 4 +4B8 3 p) =0, =5(135 4 - 4A 2 p 4) =0, P yield by elimination of S and p the discriminant D =64B-A2.

00Let a covariant of degree e in the variables, and of degree 8 in the coefficients (the weight of the leading coefficient being w and n8-2w = Ã¯¿½), be Coxl -}- ec l l 1 x 2 -{-...

00- zn +9 1 -z2.1 -z3....1-z8; and since this expression is unaltered by the interchange of n and B we prove Hermite's Law of Reciprocity, which states that the asyzygetic forms of degree 0 for the /t ie are equinumerous with those of degree n for the The degree of the covariant in the variables is e=nO-2w; consequently we are only concerned with positive terms in the developments and (w, 0, n) - (w - r; 0, n) will be negative unless nO It is convenient to enumerate the seminvariants of degree 0 and order e=n0-2w by a generating function; so, in the first written generating function for seminvariants, write z2 for z and az n for a;.

00A Similar Theorem Holds In The Case Of Any Number Of Binary Forms, The Mixed Seminvariants Being Derived From The Jacobians Of The Several Pairs Of Forms. If The Seminvariant Be Of Degree 0, 0' In The Coefficients, The Forms Of Orders P, Q Respectively, And The Weight W, The Degree Of The Covariant In The Variables Will Be P0 Qo' 2W =E, An Easy Generalization Of The Theorem Connected With A Single Form.

00Q 1 The Unreduced Generating Function Which Enumerates The Covariants Of Degrees 0, 0' In The Coefficients And Order E In The Variables.

00Taking the variables to be x, y and effecting the linear transformation x = X1X+1.11Y, y = X2X+It2Y, X 2 +Y2X Y Xl - X2 y = _ x X I + AI R X 122 so that - Ã¯¿½l b it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence.

00Instead of following the motion of each individual part of a material system, he showed that, if we determine its configuration by a sufficient number of variables, whose number is that of the degrees of freedom to move (there being as many equations as the system has degrees of freedom), the kinetic and potential energies of the system can be expressed in terms of these, and the differential equations of motion thence deduced by simple differentiation.

00Pierpont, Lectures on the Theory of Functions of Real Variables (1905).

00In the Eulerian notation u, v, w denote the components of the velocity q parallel to the coordinate axes at any point (x, y, z) at the time t; u, v, w are functions of x, y, z, t, the independent variables; and d is used here to denote partial differentiation with respect to any one of these four independent variables, all capable of varying one at a time.

00For further details of his mathematical investigations see the articles Theory of groups, and Functions Of Complex Variables.

00Since the two expressions (9) are the partial differential-coefficients of a single function E of the independent variables v and 0, we shall obtain the same result, namely d 2 E/d0dv, if we differentiate the first with respect to v and the second with respect to 0.

00A similar expression for the variation of the specific heat S at constant pressure is obtained from the second expression in (8), by taking p and 0 as independent variables; but it follows more directly from a consideration of the variation of the function (E+pv).

00

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