When the elements of the first row of a determinant are all positive integral powers of one quantity, the elements of the second the like positive integral powers of another quantity, and so on, the determinant is called an ALTERNANT; for example, .

Every alternant of the nth degree is evidently a function of n variables, viz., the n quantities whose powers are the elements. To interchange two of these variables would be the same as to interchange two of the rows of the determinant, and therefore would have the effect of merely changing the sign of the function. A function having this property, and therefore closely resembling a symmetric function, Cauchy called a symmetric function also distinguishing the two kinds as alternating and permanent.