EQUATION 499 terms in the second and third lines are the like functions with (a , b , c) and (a", b", c") respectively. There is an apparently arbitrary transposition of lines and columns ; the result would hold good if on the left- hand side we had written (a, /?, y), (a , J3 y ), (a", (3", y"), or what is the same thing, if on the right-hand side we had transposed the second determinant; and either of these changes would, it might be thought, increase the elegance of the form, but, for a reason which need not be explained, 1 the form actually adopted is the preferable one. To indicate the method of proof, observe that the deter minant on the left-hand side, qua linear function of its columns, may be broken up into a sum of (3 3 = ) 27 deter minants, each of which is either of some such form as a8y a , a , b a , a . b a" , a" , b where the term a/2y is not a term of the a/?y-determi- nant, and its coefficient (as a determinant with two iden tical columns) vanishes ; or else it is of a form such as a b , c b , c b" , c" that is, every term which does not vanish contains as a factor the a&c-determinant last written down ; the sum of all other factors a /? y" is the a/?y- determinant of the formula; and the final result then is, that the determinant on the left-hand side is equal to the product on the right-hand side of the formula. 7. Decomposition of a determinant into complementary determinants. Consider, for simplicity, a determinant of the fifth order, 5 = 2 + 3, and let the top two lines be a , b , c , d , e a , b , c , d , e then, if we consider how these elements enter into the determinant, it is at once seen that they enter only through the determinants of the second order a, 6 a , b , &c., which can be formed by selecting any two columns at pleasure. Moreover, representing the remaining three lines by a" , b" , c" , d" , c" of , b" , c " , d" , e " a"", b"" , c"" , d " , e"" it is further seen that the factor which multiplies the determinant formed with any two columns of the first set is the determinant of the third order formed with the com plementary three columns of the second set; and it thus appears that the determinant of the fifth order is a sum of all the products of the form , b c" , d" , e , b c " , d " , e c"", d"", c tho sign being in each case such that the sign of the term alt . c"d "e"" obtained from the diagonal elements of the component determinants may be the actual sign of this term in the determinant of the fifth order; for the product written down the sign is obviously -f . Observe that for a determinant of the n-th order, taking the decomposition to be 1 + (n- 1), we fall back upon the equations given at the commencement, in order to show the genesis of a determinant. 8. Any determinant , formed out of the elements of a, b the original determinant, by selecting the lines and columns at pleasure, is termed a minor of the original determinant ; and when the number of lines and columns, or order of the determinant, is n 1, then such determinant is called a first minor ; the number of the first minors is = n 2 , the first minors, in fact, corresponding to the several elements of the determinant that is, the coefficient therein of any term whatever is the corresponding first minor. The first minors, each divided by the determinant itself, form a system of elements inverse to the elements of the determinant. A determinant is symmetrical when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other ; if they are equal and opposite (that is, if the sum of the two elements be =0), this relation not extending to the diagonal elements themselves, which re main arbitrary, then the determinant is slieio ; but if the relation does extend to the diagonal terms (that is, if these are each = 0), then the determinant is skeiv symmetrical ; thus the determinants a , h , g ]h,b,f ,f,c v , -p. b , A A., c , v ,-fj. -v , , A H , - A, 1 The reason is the connexion with the corresponding theorem for the multiplication of two matrices. are respectively symmetrical, skew, and skew symmetrical. The theory admits of very extensive algebraic develop ments, and applications in algebraical geometry and other parts of mathematics ; but the fundamental properties of the functions may fairly be considered as included in what precedes. THEOKY OF EQUATIONS. 9. In the subject "Theory of Equations" the term equation is used to denote an equation of the form x n -p i x n ~ l . . . p n = Q, where p v p 2 . . . p n are regarded as known, and x as a quantity to be determined : for short ness the equation is written f(x) = . The equation may be numerical; that is, the coeffi cients p v pi . . p n are then numbers, understanding by number a quantity of the form a + fii (a. and /? having any positive or negative real values whatever, or say each of these is regarded as susceptible of continuous variation from an indefinitely large negative to an indefinitely large positive value), and i denoting *J~ . Or the equation may be algebraical ; that is, the coeffi cients are not then restricted to denote, or are not explicitly considered as denoting, numbers. I. We consider first numerical equations. (Real theory, 10 to 14 ; Imaginary theory, 15 to 18.) 10. Postponing all consideration of imaginaries, we take in the first instance the coefficients to be real, and attend only to the real roots (if any); that is, p v p*,...p n are real positive or negative quantities, and a root a, if it exists, is a positive or negative quantity such that a n -p l a n ~ l ... p rt = 0, or say, /() = (). The fundamental theorems are given under ALGEBRA, sections x., xiii., xiv.; but there are various points in the theory which require further development. It is very useful to consider the curve y=f(* or, what would come to the same, the curve A.y=f(x), but it is better to retain the first-mentioned form of equation, drawing, if need be, the ordinatey on a reduced scale. For instance, if the given equation be x 3 - 6x~ + 1 IT - G OC = 0,- then the curve y = x* - Gx 2 +llx - 606 is as shown in the figure at page 501, without any reduction of scale for the ordinate. It is clear that in general y is a continuous one-valued function of x, finite for every finite value of x, but becom ing infinite when x is infinite ; i.e., assuming throughout that the coefficient of x" is + 1, then when a-=o>. y = + oo ; but when x= - o> , then y = + or - GO , - The coefficients were selected so that the roots might be nearly
1, 2, 3.