## Equations Sentence Examples

- Energyequations, such as the above, may be operated with precisely as if they were algebraic
**equations**, a property which is of great advantage in calculation. - 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. - But the bulk of the work consists of problems leading to indeterminate
**equations**of the second degree, and these universally take the form that one or two (and never more) linear or quadratic functions of one variable x are to be made rational square numbers by finding a suitable value for x. - A few problems lead to indeterminate
**equations**of the third and fourth degrees, an easy indeterminate equation of the sixth degree being also found. - Often assumptions are made which lead to
**equations**in x which cannot be solved "rationally," i.e. - Since the distance of a body from the observer cannot be observed directly, but only the right ascension and declination, calling these a and 6 we conceive ideal
**equations**of the form a = f (a, b, c, e, f, g, t) and 5=0 (a, b, c, e, f, g, t), the symbols a, b,. - If the values ofa and 6, defining the position of the body on the celestial sphere, are observed at three different times, we may conceive six
**equations**like the above, one for each of the three observed values of a and S. - Then by solving these
**equations**, regarding the six elements as unknown quantities, the values of the latter may be computed. - For Tartaglia's discovery of the solution of cubic
**equations**, and his contests with Antonio Marie Floridas, see Algebra (History). - His largest work,Trattato generale di numeri e misure, is a comprehensive mathematical treatise, including arithmetic, geometry, mensuration, and algebra as far as quadratic
**equations**(Venice, 1556, 1560). - This section treats of such subjects as nomenclature, formulae, chemical
**equations**, chemical change and similar subjects. - The distribution of weight in chemical change is readily expressed in the form of
**equations**by the aid of these symbols; the equation 2HC1+Zn =ZnCl2+H2, for example, is to be read as meaning that from 73 parts of hydrochloric acid and 65 parts of zinc, 136 parts of zinc chloride and 2 parts of hydrogen are produced. - The combination, as it is ordinarily termed, of chlorine with hydrogen, and the displacement of iodine in potassium iodide by the action of chlorine, may be cited as examples; if these reactions are represented, as such reactions very commonly are, by
**equations**which merely express the relative weights of the bodies which enter into reaction, and of the products, thus Cl = HC1 Hydrogen. - They appear to differ in character; but if they are correctly represented by molecular
**equations**, or**equations**which express the relative number of molecules which enter into reaction and which result from the reaction, it will be obvious that the character of the reaction is substantially the same in both cases, and that both are instances of the occurrence of what is ordinarily termed double decomposition H2 + C12 = 2HC1 Hydrogen. - A physicist, however, does more than merely quantitatively determine specific properties of matter; he endeavours to establish mathematical laws which co-ordinate his observations, and in many cases the
**equations**expressing such laws contain functions or terms which pertain solely to the chemical composition of matter. - Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic
**Equations**," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. - 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**. - 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**. - 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. - Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential
**Equations**to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces. - They teach further the solution of problems leading to
**equations**of the first and second degree, to determinate and indeterminate**equations**, not by single and double position only, but by real algebra, proved by means of geometric constructions, and including the use of letters as symbols for known numbers, the unknown quantity being called res and its square census. - In the Flos
**equations**with negative values of the unknown quantity are also to be met with, and Leonardo perfectly understands the meaning of these negative solutions. - Jahn, 2 the processes at the anode can be represented by the
**equations**2CH 3 000+H 2 0 =2CH3 000H+0 2 C H 3. - The difference of potential between two solutions of a substance at different concentrations can be calculated from the
**equations**used to give the diffusion constants. - The results give
**equations**of the same logarithmic form as those obtained in a somewhat different manner in the theory of concentration cells described above, and have been verified by experiment. - On these lines the
**equations**of concentration cells, deduced above on less hypothetical grounds, may be regained. - The adjoint determinant will be seen subsequently to present itself in the theory of linear
**equations**and in the theory of linear transformation. - Let there be 2n
**equations**r }}? - Linear
**Equations**.-It is of importance to study the application of the theory of determinants to the solution of a system of linear**equations**. - Suppose given the n
**equations**fl= = allxl +a12x2 + ï¿½ ï¿½ ï¿½ + annxn = 0, f2 =a21x1+a22x2+ï¿½ï¿½ï¿½+a2nxn =0, fn =anlxl +an2x2+ï¿½ï¿½ï¿½ +annxn = 0. - Denote by A the determinant (a11a22ï¿½ï¿½ï¿½ann)ï¿½ Multiplying the
**equations**by the minors A l, .., A2,,,,ï¿½ï¿½ï¿½Ani., respectively, and adding, we obtain x 1 (ai, Aig+a2p.A2lc+ï¿½ï¿½ï¿½+anï¿½Anï¿½) =xï¿½A=o, since from results already given the remaining coefficients of x 11' x 2, ...x ï¿½ 'i xï¿½+I,...x, vanish identically. - =xï¿½ = o is the only solution; but if A vanishes the
**equations**can be satisfied by a system of values other than zeros. - For in this case the n
**equations**are not independent since identically Alï¿½ft+ A2ï¿½ f2+...+Anï¿½fn = 0, and assuming that the minors do not all vanish the satisfaction of ni of the**equations**implies the satisfaction of the nth. - Consider then the system of ni
**equations**a21xi+a22x2+ï¿½ï¿½ï¿½ + a2nxn = 0 a31x1+a32x2+ï¿½ï¿½ï¿½+a3nx,, =0 an1x1 + an2x2 + ï¿½ ï¿½ ï¿½ +annxn = 0, which becomes on writing xs = y 8, a21y1+ a 22y2 + ï¿½ ï¿½ ï¿½ + a 2,n-lyn-1 + a 2n = 0 a3lyl +a32y2+ï¿½ï¿½ï¿½ +a3,n-lyn-i+a3n =0 an1 y1 +an2y2 +ï¿½ï¿½ï¿½ +an, n-lyn-1 +ann = 0. - We can solve these, assuming them independent, for the - i ratios yl, y2,...yn-iï¿½ Now a21A11 +a22Al2 ï¿½ ï¿½ ï¿½ = 0 a31A11+a32Al2 +ï¿½ ï¿½ï¿½ +a3nAln = 0 an1Al1+an2Al2 +ï¿½ï¿½ï¿½+annAln =0, and therefore, by comparison with the given
**equations**, x i = pA11, where p is an arbitrary factor which remains constant as i varies. - For further information concerning the compatibility and independence of a system of linear
**equations**, see Gordon, Vorlesungen fiber Invariantentheorie, Bd. - 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. - R is a function of the coefficients which is called the " resultant " or " eliminant " of the k
**equations**, and the process by which it is obtained is termed " elimination." - We cannot combine the
**equations**so as to eliminate the variables unless on the supposition that the**equations**are simultaneous, i.e. - Each 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. - Consider two binary
**equations**of orders m and n respectively expressed' in non-homogeneous form, viz. - 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.