EQUATION. 164 elaborate system of symbolism. The equations of VI Kalsadi (fifteenth century; are models of brev- ity and this plan for solving linear equati modified Hindu method, was what was later known as the regula falsi. See False Position. The Europeans of the .Middle Ages made little advance in the solving of equations until the discovery by Ferro, Tartaglia, and Cardan (six- century) of the general solution of the cubic equation. The solution of the biquadratic equation >oon followed, and the general quintic was attacked. But, although much was done to advance the general theory of the equation by Vandernionde, Euler, Lagrange, Bezout, Waring. Malfatti, and others, it was not until the begin- ning of the nineteenth century that equation-- of a degree higher than the fourth received satis- factory treatment. Euffini and Abel were the first to demonstrate that the solution, by alge- braic methods, of a general equation of a degree higher than the fourth is impossible, and to direct investigation into new channels. Mathe- maticians now sought to classify equations which could be solved algebraically, and to discover higher methods for those which could not. Gauss solved the cyclotomic group. Abel the group known as the Abelian equations, and Galois stated the necessary and sufficient condition for the algebraic solubility of any equation as fol- low-: If the degree of an irreducible equation is a prime number, the equation is soluble by radicals alone, provided the roots of this equa- tion can be expressed rationally in terms of any two of them. As to higher methods. Tschirn- hausen. Bring, and Hermite have shown that the general equation of the fifth degree can be put in the form f' — t — A = 0; Hermite and Kro- necker solved the equation of the fifth degree by elliptic functions; and Klein has given the sim- plest solution by transcendental functions.
few of the more important properties of
equation* are: (1) If r is a root of the equation i = 0, then x — r is a factor of / 2 being a root of ..- : + 2x — 8 = 0, then a? — 2 is a factor of :r- 4 -■ s - (2) If /"ill is divisible by as — r, r is a root of f{x) = 0; e.g. in I > — 2) (.r 5 -f X + 1) = 0, x — 2 i- a factor, hence x — 2 = satisfies the equal ion, and x = '-. (3) Every equation of the nth degree has n roots and i re (the fundamental theorem of equations due to Harriot, or, in its complete form, to D'Alembert) : e.g. a;* — 1 = has four . x= 1, — 1. i. — '. and no more. I i l li ifficients of an equation arc func- tions of ii- roots. Thus, in i°- r o 1 » n "' 4-« : a:"- J -|-....« =0 quadratic equation, taken as EQUATION. x 2 + px + q = 0, may be o ±VP 2 - 4 2 The expression if O, - i are I he roof -. i hen" VP ' — 4 2 is the discriminant, for if 4^> p' the roots are complex; if iq = p" the roots are equal ; if 4<jr < p- the roots are real; and if p ! — iq is a perfect square, the roots are rational. Similarly the dis- criminant of the cubic x 3 + 3hx + g = is The discriminants of equations of higher degree are fully explained in works on the theory of equations. A differential equation is an equation involv- ing differential coefficients (see Calculus) ; e.g. ^3L+a^S.= x, dx 3 clx from which it is required to find the relation be- tween y and x. The theory of the solution of such equations is an extension of the integral calculus, and is a branch of study of the highest importance. For the general theory of equations, consult : Burnside and Panton, Theory of Equations (4th ed., London. 1899-1901), the appendix to which contains valuable historical material; Peterson, Theorie des equations algebriques (by Laurent, Paris, 1897); Salmon, Lessons Introductory hi Modern Higher Algebra (Dublin, 1859, ami subsequent editions) ; Serret, Cours d'algeore supirieure (3d ed.. Paris. 1860 i ; Jordan, Traite des substitutions et des equations algeTyriques ( Paris, 1870). An extensive work, covering both history and method, is Matthiessen, Grundziig der antiken und modernen Algebra der literal n Gleichungen (Leipzig, 1896). EQUATION, Annual. One of the most con spicuous of the subordinate fluctuations in the in is motion., due to the action of the sun. It consists in an alternate increase and decrease in the moon's longitude, corresponding with the earth's situation in its annual orbit, i.e. to its angular distance from the perihelion, and there- fore ii has a year instead of a month, or aliquot part of a month, for its period. EQUATION, Chemical. See Chemistry. EQUATION, Personal. A very important factor in astronomical observations. Two observ- ers, each of admitted skill, often differ in their record of the same event — as the passage of n star before the wires of a transit in- strument — by a quantity nearly the same This i 1 n-l i I (5) The number of positive roots ,,f /i.r) =0 for all observations by those persons. , a, = r, r„ + r, r ; , + quantity is their relative personal .equation. Each _ /, ", , ,- ,- J- i observer habitually notes the time too early or , " :i — V ' i ' l ' 3 I 'i'a'4 i ■ • • n . •, , , . .„ ,.__ to,, late, by a small and nearly uniform portion of i second. This quantity is his absolute per- sonal equation. Machines have been invented for determining the amount of personal equation by reproducing artificially (he kind of observation, usually affected with this form of error in actual work on the sky. The so-called Repsold appa- ratus is ;i mechanical device which so changes tl indition of observation with a transit instru- ment or meridian circle that the personal equa tion is removed altogether, and its quantitative evaluation is rendered unnecessary, i he numlier of changes of signs I Ii - artes's rule of signs, i I-', g. in .r — 1 = o then a re 3 changes ',-. heme there can I"' no more than 3 |io-it I i functions associated with tin' I equation which serve to distinguish the nature of i : called discrvmi- ll form of the roots of the
