INTERMITTANT is a special form of permutant which need not be here further explained.
CumuLasir.-The name has been given to the function which is the numerator or denominator of a continued fraction. Such function may be exhibited (and indeed naturally presents itself) in the form of a determinant, thus the cumulant (a bed) or numerator of the 1 1 1 fractions a + is a, 1, • , • -1, b, 1, •• ,-1, c, 1 • , • -1, cl and so for a greater number of terms. The developed expression is a bed+ ab + ad+be+ 1, which is formed from the produced abed by successively omitting each product (cd, be, ab), or set of products (cd, ab) of two consecutive letters ; in like manner the cumulant (abcde)isabcde+abc+abe+ade+a+e+e.
DIATRIX.-The term might be used to denote any arrangement of terms, but in a restricted sense it denotes a square or reotangular arrangement of terms, and it is thus employed in the theory of determinants.
To show further how the notion of a matrix is made use of, it may be remarked that a system of linear equations l=a x+b y+c z, = a' x+b' +6/ z, C= a"x +b"y+c"z is in the notation of matrices represented by n, C)= a, b, ,a, b', c' a", b", c" The corresponding set of equations which give (x, y, z) in terms of Q, sr, C) is represented by, a, b, c ) a', b, c' I a" ,b",c" and we have thus the definition of the inverse or reciprocal matrix : it follows from the theory of determinants that the terms of the recipro cal matrix are the first minor determinants formed out of the original matrix, each of them divided by the determinant formed out of the original matrix ; but in writing down the expression some attention is required with respect to the arrangement and signs of the terms.
Similar considerations lead to the notion of multiplying or compound ing together two or more matrices. As an instance of such composition, take ) ( a, ) = Spa+ p$+ o$' PC d i I a', cr7' where it is to be observed that the lines of the first or further compo nent matrix are compounded with the columns of the second or nearer component matrix to form the lines of the compound matrix. The words further, nearer, are used in reference to a set (x, y) which is, or may be considered to be, understood at the right of each side of the equation. A matrix may be compounded with itself once or oftener,
giving rise to a positive power of such matrix ; the notion of the nega tive powers is deducible from that of the inverse or reciprocal matrix, and the same process of generalisation as is employed for powers of a single quantity leads to the notion of the fractional powers of a matrix. As a definition of addition, matrices are added together by the addi tion of their corresponding terms, and as a particular case of the multiplication or composition of matrices we have the multiplication of a matrix by a single ,quantity, effected by multiplying by such quantity each term of the matrix ; all these notions together lead to the notion of functions of a matrix.
As an instance of the employment of the notation of matrices for another purpose, take 6, c 3f, I) a, C), d, e a", V, c' used to denote the Low-linear function (a x+b y+c +(ex+b"y+e': C which includes a, h,9 ) (x, 9, f, used to denote the quadric function z+2grx +2 Azy.
The last preceding notation is an instance of a symmetrical matrix: the terms skew, skew symmetrical, already explained with respect to determinants, apply also to matrices.
Resutrerr.—If there be a system of equations between the same number of unknown quantities (it is assumed that the several equations are of the form u=o, where u is a rational and integral homogeneous function), then the function of the coefficients which equated to zero expresses the result of the elimination of the un known quantities from the several equations, or (what is the same thing) gives the condition for the existence of a set of values satis fying the equations simultaneously—is the Resultant of the equations, or of the functions which are thereby put equal to zero. In the case of two (non-homogeneous) equations involving a single unknown quantity, we may say more briefly that the resultant is the func tion which equated to zero gives the condition for the existence of a common root. In the particular case of a system of linear equations between as many unknown quantities, the resultant is the determinant formed with the coefficients of the equations.