Variables sentence example

variables
  • Certain clusters contain a remarkable number of short-period variables.
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  • Thus the cluster Messier 5 was found at Harvard to contain 185 variables out of 900 stars examined.
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  • Moreover, 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.
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  • Stars 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.
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  • A 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.
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  • The phenomena of long-period variables show that the surface brilliancy may vary very greatly, even in the same star.
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  • For 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.
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  • Roberts concluded in this way that the average density of the Algol variables and their eclipsing companions is about one-eighth that of the sun.
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  • 4, together constitute a catalogue of 3734 variable stars; ephemerides of over 800 variables are given in the Vierteljahrsschrift of the Astronomische Gesellschaft.
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  • The 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.
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  • These 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.
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  • For 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.
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  • To 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.
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  • If 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).
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  • Secondly, 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).
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  • This 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.
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  • We 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'.
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  • Now, 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.
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  • Since 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.
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  • In 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.
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  • This 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.
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  • Functions of the variables possessing this property of remaining constant are termed integrals.
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  • The 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.
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  • When 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).
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  • In 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.
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  • In 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.
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  • When 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.
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  • A further release of data, which contains additional derived variables, will be made available at a later date.
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  • The counterexample is the quantum exterior algebra in 2 variables.
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  • The higher the number of variables the longer the optimization algorithms will take to run.
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  • Three TMD related symptoms and reported bruxism were used as dependent variables.
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  • Sets of variables engage in health insurance provider colorado some.
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  • This paper proposes a new test of independence based on the maximum canonical correlation between pairs of discrete variables.
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  • Mantel statistics may turn out significant even if they have values that would seem very low for a Pearson product-moment correlation between two variables.
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  • Main outcome measures: Relationship between variables examined by calculating Pearson's product moment correlation coefficients.
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  • The main dependent variables were levels of fear and worry in prisoners and officers.
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  • This in turn will simplify the derivation of design principles based on these variables.
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  • You can examine whether or not variables are normally distributed with histograms of frequency distributions.
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  • The good news is that Perl doesn't always use a dollar sign to identify variables.
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  • Donald script in eden, it is necessary to associate display actions with variables.
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  • See ENVIRONMENT VARIABLES for detailed descriptions of the mailx variables. sh ell [shell-command] !
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  • These variables are now used to dynamically generate the SiteMap.
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  • The two main independent variables were the prison wing and the category of offense.
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  • The expression may contain variables of the form " { i } " where i is a nonnegative integer.
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  • Thus, language planning is a very complex process, involving linguistic, social, psychological, environmental and other variables.
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  • People's welfare depends upon real variables, not nominal variables.
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  • It is assumed that the data (for the variables) represent a sample from a multivariate normal distribution.
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  • Create variables, use variables in assignment statements and write code to carry out simple arithmetic operations.
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  • Group differences in average volume of ischaemic penumbra salvaged will be analyzed by t-test, adjusting for baseline differences in prognostic variables.
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  • The range for x can be further extended by using quadruple precision for the input x and related variables (see LONG WRITE-UP ).
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  • Given two new first-order variables a and b, the translations ST a and ST b are defined by mutual recursion.
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  • Bespoke languages are deemed necessary in order to avoid introducing confounding variables.
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  • You can declare variables at the start of any compound statement.
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  • We proceed as is usual for modeling a problem, by defining variables.
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  • The number of manipulated variables which may be used in control loops.
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  • These are all models that have categorical variables, as it's major components.
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  • Starting with the two dummy variables less likely to these are the.
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  • These environment variables may be used to control the executable's behavior.
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  • The integer variables LINES and COLS are defined in curses.h and will be filled in by initscr with the size of the screen.
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  • Keywords OBJ Set this keyword to free object heap variables only.
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  • When 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.
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  • Suppose n dependent variables yl, y2,���yn, each of which is a function of n independent variables x1, x2 i ���xn, so that y s = f s (x i, x 2, ...x n).
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  • � 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,���xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .
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  • The 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.
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  • In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " �� ' 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.
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  • A 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'.
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  • Let 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 = �), be Coxl -}- ec l l 1 x 2 -{-...
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  • Taking 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 - �l b it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence.
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  • In 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.
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  • He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n 2 differential coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations (see Algebraic Forms).
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  • Sensors can constantly monitor moisture levels in the soil, the size and color of the plants, air quality, nutrient levels in the soil, amount of sunlight, and hundreds of other variables.
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  • The results of these experiments show consistent effects of all the contextual variables on quantifier judgements.
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  • Fourth, the quantifier rules provide no indication as to what terms or free variables must be used in their deployment.
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  • The internal units of the variables are bohr for bond lengths and radians for angles.
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  • The detailed rationale for the selection of variables to be measured was (unfortunately) implicit rather than explicit.
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  • To implement read-only variables, command will have to restore the old value of the variable.
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  • Recourse variables would represent the liquidation (selling) of assets to meet liabilities.
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  • The program is designed so as to reconstruct the polynomial in two variables represented by this equation.
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  • Dichotomous variables persisted of adverse california health insurance rating selection retiree health benefits.
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  • Justification: lemma bounding the number of roots of a polynomial of degree d in n variables.
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  • After controlling for hospital and socioeconomic variables the excess risks were lowered.
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  • Static variables can be used to emulate constants, values that do n't change.
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  • Any subscript expression occurring in the left part variables are evaluated in sequence from left to right.
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  • Using regression analysis it was found that these psychiatric symptomatology variables were differentially associated with the four factors for both males and females.
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  • A three-position tweeter level control accommodates a range of room acoustic variables.
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  • Of course, Prolog remembers to unbind any variables exactly at the places in the tree where they were bound.
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  • That is to say, no unbound variables are allowed. names represent signals in syndi and are syntactically similar to identifiers.
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  • Do we sometimes try to universalize the process and undervalue these important variables?
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  • What you 're seeing is PHP warn you about unset variables.
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  • These are all models that have categorical variables, as it 's major components.
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  • These environment variables may be used to control the executable 's behavior.
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  • The two primary variables that affect the compound interest equation are length of time and rate of return.
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  • Although the workings of systems are complex, involving a series of 24 satellites and various calculations using different variables, the end result is really quite straightforward in that it is easy to pinpoint a position.
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  • How to buy a bicycle seems fairly straightforward but in reality, there are quite a few variables out there to consider before plopping down the cash.
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  • Even though a seven-year-old cat may be the equivalent of a middle-aged human, there are a lot of other variables that can affect the lifespan of the animal.
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  • You can expect to pay $400 to $700 or more depending on the breeder, the location, bloodlines, color and other variables.
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  • The website Smart Money.com has a few calculators and worksheets that enable you to plug in certain variables to determine debt status.
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  • The basis for spousal support relies on many different variables, making it impossible to create one specific calculation.
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  • In addition to atmospheric variables, there are also zoning restrictions to consider.
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  • Even though space is at a premium in most dorms, all it takes is juggling a few variables to have the kind of room in which you can comfortably get a good night's rest, hang with friends or who knows, maybe even crack a book now and again.
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  • Witches are such an artful and eclectic costume choice since there are so many variables of witch styles and their differing personalities.
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  • You do need to type your equation into a text box, giving Encalc any variables needed.
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  • Let's take a look at some of the variables that determine how to choose the right pot for the job.
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  • This is due to the many variables including a person's height, boot length, type of boot and skill level.
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  • Each person has different talents and preferences, and those variables impact how stressful a particular job is for any given individual.
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  • While these variables can help make a love poem sound sweet, most poems will sound sweet to the person because of the thoughtfulness you put into it.
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  • You can shop the site by size, by style, by collection, by price, and many other variables.
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  • Most everyone has heard that cigarette smoking or tobacco chewing and cancer are corresponding variables with poor health and a shorter life span; yet are they really?
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  • Additionally there are ever changing promotions by region, state, agency, age, occupation, and many other variables that can affect the price.
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  • There are personal preferences to take into account, as well as variables such as price and quality.
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  • By changing variables within the calculator, the homeowner will be able to determine some possible forms for the home addition, and later discuss these ideas with an architect.
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  • Any wine enthusiast will tell you that there are many variables to think about when building a home wine cellar, including temperature, storage systems, labeling, preservation, and more.
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  • Hair color will last up to three months for most people, depending on how often they wash their hair, the amount of sun they get and other variables.
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  • Applying for a reverse mortgage is a big decision, and there are many variables involved.
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  • There are many resources to help seniors understand the many variables surrounding health insurance rates for senior citizens in Ohio.
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  • Some use real world historical battles, allowing players to change variables to see if they could have changed the outcomes; others use original war situations, allowing the player to wage unreal campaigns.
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  • Your moment-to-moment goal is simple: eliminate the variables.
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  • There are too many variables and he will be unable to evaluate the condition of the book.
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  • There are actually hundreds of variables that determine the value of antiques.
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  • There are many variables when it comes to the value of any retired piece.
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  • Other variables that might affect the value of a book include whether the book is a first edition copy, a reprint or a book club edition.
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  • However, the entire set of variables used to determine how much an antique is worth varies somewhat based on the type of antique being valued.
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  • Typical performance, depending on a number of variables, is probably closer to the 1Mbps to 2Mbps range.
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  • In other cases, the diagnosis remains uncertain because of the mildness of the child's symptoms, the absence of a family history of the syndrome, and other variables.
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  • Also called insulin-like growth factor (IGF-1), somatomedin C is considered a more accurate reflection of the blood concentration of hGH because such variables as time of day, activity levels, or diet do not influence test results.
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  • The CAT's creators suggest a series of ten variables to consider when interpreting the results.
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  • These variables include the story's major theme, the major character's needs, drives, anxieties, conflicts, fears, and the child's conception of the external world.
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  • New variables that are progressively more difficult are gradually added.
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  • Assessment is a process of gathering and documenting information about the achievement, skills, abilities, and personality variables of an individual.
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  • If no fecal impaction is found, the pediatrician works with a counselor or psychiatrist to analyze the variables that characterize the encopresis.
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  • Academic statistics can tell you a variety of things by measuring different variables that affect academic achievement.
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  • Looking at factors such as parents' education levels and other variables, NHERI compared students' academic achievement to determine what made a significant difference (if anything) in students' performance.
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  • With more options than many online calculators, Karl's gives you a chance to input an assortment of variables and see the results based on your information.
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  • The amount of debt prospective homebuyers carry affects the three primary variables that determine interest rate on the loans they will receive.
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  • Adding more variables to an already volatile situation seldom solves anything.
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  • With all of these variables, it is difficult for sperm to fertilize an egg and, if it does, the egg most likely will not implant in the uterus.
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  • Even if you have sex during your most fertile time, this doesn't necessarily mean you'll get pregnant because there are other variables besides ovulation to consider.
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  • Choosing the right birth control method requires carefully considering many variables.
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  • While this is a more accurate amount based on size differencials, there are other variables that come into play as to how much you may need.
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  • There are many variables to consider when evaluating the right unit for your home.
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  • If your reading does not appear to be accurate, think about the variables of its meaning and how they may be more accurate.
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  • It would be fool hardy to say that the effects of single parenting are always negative when there are so many other variables to consider.
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  • There are just too many variables surrounding birth order to have the controls necessary for a valid peer-reviewed study.
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  • Variables include things like socioeconomic status, skills of the parents and education, which vary widely from subject to subject.
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  • Too many other variables factor into the development of personality for birth order to hold the sole responsibility for personality.
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  • Scientists from Notre-Dame examined the effect of school uniforms in Miami-Dade county in Florida, taking into account other variables such as economic class, race, and more.
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  • The website Smart Money has calculators and worksheets that allow you to determine a debt status by plugging in certain variables.
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  • That way, if you have to toss out variables, you'll know that you can ease your budget by changing length, style, and manufacturer.
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  • Henna tattoos could last up to four weeks, but several variables, like temperature, placement, and care come into play.
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  • The number of variables fluctuates by website, but basic features allow travelers to take side trips and find attractions or shopping along the designated route.
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  • There are many variables to consider, such as color and style.
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  • It focuses on a relationship between two or more variables in a controlled environment.
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  • You can play with the numbers to see how these variables affect one another, and this will help you settle on your car budget.
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  • It's difficult to predict the average life span of a brake pad because typical wear is dependent on several variables.
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  • A payment calculator is a tool you can use to input certain variables to determine your monthly car payment.
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  • Obviously, with all of these variables, it's not always easy to pin down an exact measurement of your car's actual fuel mileage performance during any one particular trip.
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  • This figure includes variables, such as deductibles, coinsurance and maximum coverage levels available under a policy.
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  • All of these variables can have a significant effect on the cost of your insurance, so don't completely trust the results which you receive from the online calculators.
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  • There are so many variables involved that it is far beyond the scope of this article.
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  • Given the number of different hosting companies that have reseller programs, choosing the right one is a matter of weighing variables and services offered in terms of the cost and value-added.
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  • A real ASP site would use variables that would change based on user input or other factors, using the power of ASP to create a dynamic and responsive site.
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  • Designing websites for fun or business can be a very difficult process with a lot of variables to consider, but you can greatly simplify your efforts with an easy to use web page builder.
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  • When you're ready to start doing more dynamic processing, you'll want to start assigning values to PHP variables as your page loads, and then doing something with those values.
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  • The important thing to understand here is that all variables are preceded with the "$" symbol, and secondly, this ability allows you to perform a variety of powerful tasks that will make your webpage extremely functional and dynamic.
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  • There are also what are called "predefined variables" that you can use in your PHP script without worrying about loading them with information.
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  • These variables can provide your script with real information about the web server, your user's session variables, HTTP cookies and more.
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  • The .php file gathers whatever variables the programmer wants and then generates the HTML code that is actually sent out to the user's computer screen.
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  • It is designed to take in variables, manipulate them, and then output the results.
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  • This is a very simplified example of the way this tool can be used - the variables can reveal much more than simply what people are looking for in a particular area.
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  • Instead, they should be replaced with variables.
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  • This system is of much service in following out mathematical, physical and chemical problems in which it is necessary to represent four variables.
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  • It 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.
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  • Under 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.
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  • The 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.
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  • � 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,���xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k .
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  • Schlafli 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.
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  • If 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.
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  • In 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.
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  • If, however, F involve as well the variables, viz.
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  • Invariantive forms will be found to be homogeneous functions alike of the coefficients and of the variables.
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  • There 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.
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  • All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.
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  • Every 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),...
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  • In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " �� ' 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.
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  • The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation of the variables.
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  • The 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.
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  • Q 1 The Unreduced Generating Function Which Enumerates The Covariants Of Degrees 0, 0' In The Coefficients And Order E In The Variables.
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  • Pierpont, Lectures on the Theory of Functions of Real Variables (1905).
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  • For further details of his mathematical investigations see the articles Theory of groups, and Functions Of Complex Variables.
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  • The 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.
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  • The phenomena seem to be dependent on variables such as time, and are more complicated than seemed likely at first.
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  • The first nine variables recognized in each constellation are denoted by single letters, after which combinations RR, RS, &c., are used.
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  • This was attempted variables.
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  • Besides the shortness of the period these variables possess other characteristics which differentiate them from the long-period variables.
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  • In 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.
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  • This 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.
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  • About 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.
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  • When the two stars are of equal brilliancy the minima are equal; this is the case in variables of the Geminorum type.
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  • A 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.
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  • Case of Three Variables.-In the next place we consider the resultants of three homogeneous polynomials in three variables.
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  • Similarly, if a form in k variables be expressible as a quadratic function of k -1, linear functions X1, X2, ...
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  • Let 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 = �), be Coxl -}- ec l l 1 x 2 -{-...
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  • - 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;.
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  • Instead 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.
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  • Variables of the Algol class are rendered difficult to discover by the incidental character of their fluctuations.
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  • To 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.
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  • Thus we shall be able to specify the system completely when the number of variables, viz.
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  • There are several wellmarked varieties of short-period variables; the most important are typified by the stars Algol, # Lyrae, Geminorum and S Cephei.
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  • For 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.
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  • Suppose n dependent variables yl, y2,���yn, each of which is a function of n independent variables x1, x2 i ���xn, so that y s = f s (x i, x 2, ...x n).
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  • We cannot combine the equations so as to eliminate the variables unless on the supposition that the equations are simultaneous, i.e.
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  • Resultant 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.
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  • The general theory of the resultant of k homogeneous equations in k variables presents no further difficulties when viewed in this manner.
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  • Discriminants.-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.
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  • A 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.
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  • If 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, ...
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  • Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.
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  • Stars 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.
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  • The 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.
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  • Among 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.
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  • If 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.
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  • This 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.
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  • Resultants.-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.
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  • Moreover, instead of having one pair of variables x i, x2 we may have several pairs yl, y2; z i, z2;...
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  • It is almost impossible to execute a pure controlled study of anything relating to nutrition because there are simply too many variables to consider.
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  • Computers, especially computers of the future, will have no trouble handling all the variables that influence nutrition, though there will be millions of them.
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  • Turner 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.
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  • To 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.
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  • Since 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.
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  • Taking 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 - �l b it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence.
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  • A 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).
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  • We 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).
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