E Q U E Q U 497 usually acicular in shape. It can be obtained also in crystals of the monoclinic system. It is very soluble, one part dissolving in O79 parts of water at 18 75C., and has a bitter, saline, and cooling taste. The salt is pre pared on the large scale by several methods, e.g., by the treatment of the bittern of salt works with sulphuric acid or ferrous sulphate, by which the magnesium chloride of the liquid is couverted into sulphate ; by acting on mag- nesite, the native magnesium carbonate, or on magnesian limestone, with sulphuric acid, preferably, in the case of the latter substance, after the removal of the calcium car bonate by means of hydrochloric acid; and, as in the neighbourhood of Genoa, by the roasting of pyritous ser pentine, subsequent exposure to the air and lixiviation, peroxiclization of ferrous salts by chlorine, precipitation of ferric oxide by burnt lime or dolomite, and evaporation of of the resultant solution of magnesium sulphate. The mineral waters of Seidlitz, Saidscbiitz, Pu llna, and of other places besides Epsom owe their potency to magnesium sulphate. The salt occurs in fibrous crusts or botryoidal masses in some limestone caves ; in gypsum quarries, as a result of the action of the gypsum on magnesian lime stone ; and in the old workings of mines, where it is pro duced by the oxidation of pyrites in the presence of mag nesium compounds. As a hydragogue purgative, it is in common use ; it is more especially valuable in febrile diseases, in congestion of the portal system, and in the obstinate constipation of painter s colic. To produce diuresis, the drug is far less frequently resorted to. It possesses the advantage of exercising but little irritant effect upon the bowels. In some cases, where full doses have failed, the repeated administration of small quantities has been found effectual. The chief application of Epsom salts or "Epsoms" is for weighting cotton-cloth. As a manure, magnesium sulphate has been chiefly employed as a top-dressing for clover-hay. The chlorides of mag nesium and sodium and salts of iron and of calcium may occur as impurities in Epsom salts. EQUATION. The present article includes DETERMI NANT and THEORY OF EQUATIONS ; and it may be proper to explain the relation to each other of the two subjects. Theory of Equations is used in its ordinary conventional sense to denote the theory of a single equation of any order in one unknown quantity ; that is, it does not in clude the theory of a system or systems of equations of any order between any number of unknown quantities. Such systems occur very frequently in analytical geometry and other parts of mathematics, but they are hardly as yet the subject-matter of a distinct theory ; and even Elimination, the transition-process for passing fiom a system of any number of equations involving the same number of unknown quantities to a single equation in one unknown quantity, hardly belongs to the Theory of Equations in the above restricted sense. But there is one case of a system of equations which precedes the Theory of Equations, and indeed presents itself at the outset of algebra, that of a system of simple (or linear) equations, Such a system gives rise to the function called a Determinant, and it is by means of these functions that the solution of the equa tions is effected. We have thus the subject Determinant as nearly equivalent to (but somewhat more extensive than) that of a system of linear equations ; and we have the other subject, Theory of Equations, used in the restricted sense above referred to, and as not including Elimination. DETERMINANT. 1. A sketch of the history of determinants is given under ALGEBRA; it thereby appears that the algebraical function called a determinant presents itself in the solu tion of a system of simple equations, and we have herein a natural source of the theory. Thus, considering tha equations ax + by + cz-d, a x + b y + c z d , a"x + l"y + c"s d", and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = 0, and the whole coefficient of 2 becomes = ; the factors in question are b c" - b"c, b"c-bc", be - b c (values which, as at once seen, have the desired property); we thus obtain an equation which con tains on the left-hand side only a multiple of x, and on the right-hand side a constant term ; the coefficient of x has the value a(Vc" - l"c ) 4- a (V c - Ic") + a (be - I c) , and this function, represented in the form I a , b , c , la , b , c a", b", c" is said to be a determinant ; or, the number of elements being 3 2 , it is called a determinant of the third order. It is to be* noticed that the resulting equation is a , 6 , c x= Id, b , c a , V , c d , I , c a", b", c" ; ; d", b", c" where the expression on the right-hand side is the like function with d, d , d" in place of a, a , a" respectively, and is of course also a determinant. Moreover, the func tions b c" - b"c , b"c - be", be - b c used in the process are themselves the determinants of the second order b , c b", c" We have herein the suggestion of the rule for the deriva tion of the determinants of the orders 1, 2, 3, 4, &c., each from the preceding one, viz., we have + a"b , c , b , c + a b", - a lb" . c" , d" + a"b ", c ", d "^ - a "b ,c ,d b " ,c ",d " b ,c ,d j lb ,c d b ,c ,d lb ,c ,d I H .r! A" {b",c",d and so on, the terms being all + for a determinant of an odd order, but alternately + and - for a determinant of an even order. 2. It is easy, by induction, to arrive at the general results : A determinant of the order n is the sum of the 1.2.3...?j products which can be formed with n elements out of n~ elements arranged in the form of a square, no two of the n elements being in the same line or in the same column, and each such product having the coefficient unity. The products in question may be obtained by permuting in every possible manner the columns (or the lines) of the determinant, and then taking for the factors the n elements in the dexter diagonal. And we thence derive the rule for the signs, viz., considering the primitive arrangement of the columns as positive, then an arrangement obtained therefrom by a single interchange (inversion, or derange ment) of two columns is regarded as negative; and so in general an arrangement is positive or negative according as it is derived from the primitive arrangement by an even or an odd number of interchanges. [This implies the
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