ALGEBRAIC FORMS. An algebraic form is a polynomial, all of whose terms are of the same degree in two or more variables. Thus axe+ by +c is not a form, but of order two in three variables x, y, z is a quadratic ternary form. A form is also called a quantic, this general name resembling cubic, quartic or quintic. If its order is n, it is called an n-ic. Forms in two variables are called binary; in four variables, quaternary. Forms are interesting principally because they occur in equations for unknown quantities, and, since the time of Descartes (1637), in equations of geometric loci, such as curves, surfaces, families of lines or systems of circles. Their homogeneity is closely related to the modern development of projective geometry, in which the region known as "infinity" in the older geometry has no special treatment, but is like an extra axis or plane of co ordinates, where the ratio of two co-ordinates becomes infinite or zero. Ratios take the place of measured distances. To equa tions in one unknown there will correspond binary forms; to those in two metric variables, ternary forms, and so on. Further, just as in geometry one considers often not single loci, but simply or doubly infinite systems of curves or surfaces, so algebraic forms may contain more than one set of variables; and the sets need not all occur to the same order, nor need two sets be com posed of equal numbers of variables. A problem may arise, too, calling for the use of two, three or more forms at the same time; these are then termed a simultaneous system.