# Variables Sentence Examples

- Roberts's list of southern
**variables**(ibid. - Pierpont, Lectures on the Theory of Functions of Real
**Variables**(1905). - Symbolic 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,... - Thus we shall be able to specify the system completely when the number of
**variables**, viz. - 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. - 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 = ï¿½), be Coxl -}- ec l l 1 x 2 -{-... - Suppose 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). - 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. - Case of Three
**Variables**.-In the next place we consider the resultants of three homogeneous polynomials in three**variables**. - CY 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**. - 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. - If 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. - In 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. - 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. - 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, ... - 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. - The repetition of the process brought the same results.
- - 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. - He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point
**variables-and**of u 8, u 2, u3, the contragredient line**variables-it**is completely determinate if its leading coefficient be known. - V., 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. - - 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;. - 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. **Variables**of the Algol class are rendered difficult to discover by the incidental character of their fluctuations.- Corum, 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. - 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. - 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. - Besides the shortness of the period these
**variables**possess other characteristics which differentiate them from the long-period**variables**. - There are several wellmarked varieties of short-period
**variables**; the most important are typified by the stars Algol, # Lyrae, Geminorum and S Cephei. - 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. - 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. - 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. - 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. - ï¿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,ï¿½ï¿½ï¿½xn), and we may consider the three determinants which i s 7xk, the partial differential coefficient of z i, with regard to k . - 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. - We cannot combine the equations so as to eliminate the
**variables**unless on the supposition that the equations are simultaneous, i.e. - The general theory of the resultant of k homogeneous equations in k
**variables**presents no further difficulties when viewed in this manner. - 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**. - Similarly, if a form in k
**variables**be expressible as a quadratic function of k -1, linear functions X1, X2, ... - 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. - If, however, F involve as well the
**variables**, viz. - Which 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 °, a, a 2, ...; (b 0, b, b 2, ...; 1, S2),.... - Are contragredient with the d-
**variables**x, y, z, ... - Of the symbolic factors of the form are replaced by IA others in which new
**variables**y1, y2 replace the old**variables**x1, x 2 . - 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),... - It 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) 2ï¿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. - 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 " ï¿½ï¿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. - The quartic to a quadratic. The new
**variables**y1= 0 are the linear factors of 0. - Hermite expresses the quintic in a forme-type in which the constants are invariants and the
**variables**linear covariants. - A 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. - The phenomena seem to be dependent on
**variables**such as time, and are more complicated than seemed likely at first. - 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. - The first nine
**variables**recognized in each constellation are denoted by single letters, after which combinations RR, RS, &c., are used. - Moreover, instead of having one pair of
**variables**x i, x2 we may have several pairs yl, y2; z i, z2;... - It is almost impossible to execute a pure controlled study of anything relating to nutrition because there are simply too many
**variables**to consider. - Computers, especially computers of the future, will have no trouble handling all the
**variables**that influence nutrition, though there will be millions of them. - 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. - The 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. - 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. - 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 - ï¿½l b it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence. - 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). - 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).