Graphic Representation.--For the graphic representation of systems of four components, four axes may be chosen intersecting at a point like the edges of a regular octahedron (Fig. 122).[386] Along these different axes the equivalent molecular amounts of the different salts are measured. [Illustration: FIG. 122.] [Illustration: FIG. 123.] To represent a given system consisting of _x_B, _y_C, and _z_D in a given amount of water (where B, C, and D represent equivalent molecular amounts of the salts), measure off on OB and OC lengths equal to _x_ and _y_ respectively. The point of intersection _a_ (Fig. 122) represents a solution containing _x_B and _y_C (_ab_ = _x_; _ac_ = _y_). From _a_ a line _a_P is drawn parallel to OD and equal to _z_. P then represents the solution of the above composition. It is usual, however, not to employ the three-dimensional figure, but its horizontal and vertical projections. Fig. 122, if projected on the base of the octahedron, would yield a diagram such as is shown in Fig. 123. The projection of the edges of the octahedron form two axes at right angles and give rise to four quadrants similar to those employed for the representation of ternary solutions (p. 273). Here, the point _a_ represents a ternary solution saturated with respect to B and C; and _a_P, quaternary solutions in equilibrium with the same two salts as solid phases. Such a diagram represents the conditions of equilibrium only for one definite temperature, and corresponds, therefore, to the isothermal diagrams for ternary systems (p. 273). In such a diagram, since the temperature and {316} pressure are constant (vessels open to the air), a surface will represent a solution in equilibrium with only one solid phase; a line, a solution with two solid phases, and a point, one in equilibrium with three solid phases. [Illustration: FIG. 124.] Example.--As an example of the complete isothermal diagram, there may be given one representing the equilibria in the system composed of water and the reciprocal salt-pair sodium sulphate--potassium chloride for the temperature 0° (Fig. 124).[387] The amounts of the different salts are measured along the four axes, and the composition of the solution is {317} expressed in equivalent gram-molecules per 1000 gram-molecules of water.[388]