We may also impose another condition, which is that the first line of the equilibrium polygon shall have some definite direction, such as yl. In this case the ray from the point p of the force diagram must be parallel to yl; and where this line intersects the horizontal line (produced in this case), is the required position for the pole o". Draw rays from o" to 8, r, and q, continuing the equilibrium polygon by lines which are respectively parallel to these rays. As a check on the work, the last line of the equilibrium polygon which is parallel to o"q should intersect the point z'. The triangles Oh and d pn have their sides respectively parallel to each other, and the triangles are therefore similar, and their corresponding sides are proportional, and we may therefore write the equation: o'n 1111 pn : kh.
but o'n and 0"71 are the pole distances of their respective force diagrams, while kh and lh are intercepts by a vertical line through the corre sponding equilibrium polygons. The proportion is therefore a proofs in at least a special case, of the general law that the perpendicular distances from the poles to the load lines of any two force diagrams are inversely proportional to any two intercepts in the corresponding equilibrium polygons. The above proportions prove the'theorem for the intercepts hk and hl. A similar combination of proportions would prove it for any vertical intercept between y and h. The proof of this general theorem for intercepts which pass through other lines of the equilibrium polygon, is more complicated and tedious, but is equally conclusive. Therefore, if we draw a 11V vertical intercept, such as tvw, we may write out the general proportion: In this proportion, if o"n were an unknown quantity, or the position of o" were unknown, it could be readily obtained by drawing two random lines as shown in diagram c, and laying off on one of them the distance no', and on the other line the distances vw and tw. By joining v and o' in diagram c, and drawing a line from t parallel to vo', it will intersect the line no' produced, in the point o". As a check, this distance to o" should equal the distance no" in diagram b. A practical application of this case, and one that is extensively employed in arch work, is the requirement that the equilibrium polygon shall he drawn so that it shall pass through three points, of which the abut ments are two, and some other point (such as v) is the third. After obtaining a trial equilibrium polygon whose closing line passes through the points y and z', the proper position for the pole o" which shall give the equilibrium polygon that will pass through the point v, may be easily.determined by the method described above.
The process of obtaining an equilibrium polygon for parallel forces which shall pass through two given abutment points and a third intermediate point, may be still further simplified by the appli cation of another property, and without drawing two trial equilib rium polygons before we can draw the required equilibrium polygon. It may be demonstrated that if the pole distance from the pole to the load line is unchanged, all the vertical intercepts of any two equilib rium polygons drawn with these same pole distances are equal. For example, in Fig. 215, a line is drawn from o, vertically upward until it intersects the horizontal line drawn through n in the point o". This point is the pole of another equilibrium polygon whose closing line will be horizontal, because the pole lies on a horizontal line from the previously determined point n in the load line. Any vertical inter cept of this equilibrium polygon will be equal to the corresponding intercept on the first trial equilibrium polygon; therefore, in order to draw a special equilibrium polygon for a given set of vertical loads, the polygon to pass through two horizontal abutment points and a definite third point between them, we need only draw first a trial equi librium polygon, the rays in the force diagram being drawn through any point chosen as a pole. Then, if we draw a line from the trial pole which shall be parallel with the closing line of this trial equilib rium polygon, the line will intersect the load line in the point 72. Drawing a horizontal line froin the point n in the load line, we have the locus of the pole of the desired special equilibrium polygon. Then draw a vertical through the point through which the special equilib rium polygon is to pass. The vertical distance of this point above the line joining the abutments, is the required intercept of the true equilibrium polygon. The intersection of that vertical with the upper line and the closing line of the trial equilibrium polygon, is the inter cept of the trial polygon. The pole distance of the true equilibrium polygon is then obtained by the application of Equation 46, by which the pole distances are declared inversely proportional to any two corresponding intercepts of the equilibrium polygons.