## 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). - 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. - 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. - 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. - Assuming then 01 to have the coefficients B1, B2,...B,, and f l the coefficients A 1, A21...A,n, we may equate coefficients of like powers of x in the identity, and obtain m+n homogeneous linear
**equations**satisfied by the m+n quantities B1, 2, ...B n, A 1, A 2, ...A m. - Forming the resultant of these
**equations**we evidently obtain the resultant of f and 4,. - Thus to obtain the resultant of aox 3 +a i x 2 +a 2 x+a 3, 4, =box2+bix+b2 we assume the identity (Box+Bi)(aox 3 +aix 2 +a2x+a3) = (Aox 2 +Aix+ A 2) (box2+bix+b2), and derive the linear
**equations**Boa ° - Ac b o = 0, Boa t +B i ao - A 0 b 1 - A 1 bo =0, Boa t +B 1 a 1 - A0b2 - A1b1-A2b° = 0, Boa3+Bla2 - A l b 2 -A 2 b 1 =0, B 1 a 3 - A 2 b 2 =0, = = (y l, y2,...ynl `x1, x2,...xnl for brevity. - Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and
**equations**, came into existence, and also particular linear and higher transformations connected with analysis and geometry.