Equations.

Equations. The statement in symbols of the relationship of equality existing between two algebraic expressions is termed an equation. Such expressions generally contain at least one unknown quantity. Thus $x\; -\; 2\; =\; 4\; +\; 3$ is an equation denoting that if 2 be deducted from some unknown quantity denoted by $x$, the remainder will be equal to $4\; +\; 3$, that is 7; therefore the value of $x$ in this equation is evidently 7 + 2, or 9. Any equation in one variable $x$ may be reduced to a form such as $F(x)\; =\; 0$, where $F(x)$ is a function of $x$, and this may be considered as the standard form of all equations. When written fully in the most general manner, this may be said to be equivalent to the equation

This is said to be an equation in $x$, the variable involved; $a\_0,\; a\_1,\; a\_2,\; \backslash dots,\; a\_n$ are the coefficients, either numerical or algebraical quantities. Any quantity which, when substituted for $x$, reduces the left-hand side to zero—i.e. any quantity which satisfies the equation, is termed a root of the equation. The main problem in equations is that of finding all the possible roots; this done, a complete solution is obtained. The theory of equations is a most important branch of algebra.

Identical equations are those which are always true, whatever be the value of the quantities involved; conditional equations are satisfied only by certain values for $x$. Equations are usually classified according to their degree, which is defined as being the highest degree of the involved variable. The equation written above as the standard form is, for example, of the nth degree. Simple equations, or equations of the first degree, are those in which $x$ or the unknown quantity appears only in the first power; when $x$ appears in the second power, the equation is quadratic; when $x$ is in the third power, cubic, and so on. Equations of the first, second, third, fourth, fifth, &c. degrees are sometimes called linear, quadratic, cubic, quartic (or biquadratic), quintic, &c. equations. See Burnside and Panton, Theory of Equations (1881); Todhunter's treatise on the same subject (new ed. 1880); or Chrystal's Algebra, part i. chaps. 14-19. Equation in astronomy means the corrections or reductions which must be applied to observations in order either to free them from error or otherwise to reduce them to some form more suitable for purposes of calculation. For the Equation of Time, see the article DAY. Personal equation is a correction which has to be applied to astronomical (or other) observations in order to remove relative error due to some peculiar mode of observation on the part of the observer. In astronomical observatories it is the practice to find this personal equation for all the observers with reference to one single observer, and the observations are all reduced by its application, so that finally they are tabulated as if made by one observer. The phrase has passed into common language, and is often used to denote that modification which is requisite in the statements or judgments of any person who in such matters is not free from bias or idiosyncrasy. There are also equations to the centre, allowing for the difference between the place of a planet as supposed to move in a circle and its actual place in an ellipse; and equations of equinoxes, between mean and apparent equinoxes. Equation of payments is an arithmetical rule for ascertaining at what time it is equitable for a person to pay a whole debt which is due in different parts, payable at different times. Equations are constantly used in Chemistry (q.v.).