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.