Another method of representation, due to Roozeboom, consists in employing an equilateral triangle, the length of whose _side_ is made equal to unity, or one hundred; the sum of the fractional or percentage amounts of the three components being represented therefore by a side of the triangle. In this case the composition of a ternary mixture is obtained by determining, not the _perpendicular_ distance of a point P from the three sides of the triangle, but the distance in a direction _parallel_ to the sides of the triangle (Fig. 81). Conversely, in order to represent a mixture consisting of _a_, _b_, and _c_ parts of the components A, B, and C respectively, one side of the triangle, say AB, is first of all divided into ten or one {238} hundred parts; a portion, B_x_ = _a_, is then measured off, and represents the amount of A present. Similarly, a portion, A_x'_ = _b_, is measured off and represents the fractional amount of B, while the remainder, _xx'_ = _c_, represents the amount of C. From _x_ and _x'_ lines are drawn parallel to the sides of the triangle, and the point of intersection, P, represents the composition of the ternary mixture of given composition; for, as is evident from the figure, the distance of the point P from the three sides of the triangle, when measured in directions _parallel_ to the sides, is equal to _a_, _b_, and _c_ respectively. From the division marks on the side AB, it is seen that the point P in this figure also represents a mixture of 0.5 parts of A, 0.2 parts of B, and 0.3 parts of C. This gives exactly the same result as the previous method. The employment of a right-angled isosceles triangle has also been suggested,[319] but is not in general use. [Illustration: FIG. 81.] In employing the triangular diagram, it will be of use to note a property of the equilateral triangle. A line drawn from one corner of the triangle to the opposite side, represents the composition of all mixtures in which the _relative_ amounts of two of the components remain unchanged. Thus, as Fig. 82 shows, if the component C is added to a mixture x, in which A and B are present in the proportions of _a_ : _b_, a mixture _x'_, which is thereby obtained, also contains A and B in the ratio _a_ : b. For the two triangles AC_x_ and BC_x_ are similar to the two triangles HC_x'_ and KC_x'_; and, {239} therefore, A_x_ : B_x_ = H_x'_ : K_x'_. But A_x_ = D_x_ and B_x_ = E_x_; further H_x'_ = F_x'_ and K_x'_ = G_x'_. Therefore, D_x_ : E_x_ = F_x'_ : G_x'_ = _b_ : a. At all points on the line C_x_, therefore, the ratio of A to B is the same. [Illustration: FIG. 82.] [Illustration: FIG. 83.] If it is desired to represent at the same time the change of another independent variable, _e.g._ temperature, this can be done by measuring the latter along axes drawn perpendicular to the corners of the triangle. In this way a right prism (Fig. 83) is obtained, and each section of this cut parallel to the base represents therefore an _isothermal surface_.