TO THE DETERMINATION OF THE EFFICIENCY OF MACHINERY. 31 
as would be produced by the effort in e acting on the quadrilateral frame abcd, 
loaded at the joints in this manner. This stress in fis found by drawing the 
fig. 35a, beginning with the polygon d, Jz, 1a, /,, a. The directions of all these 
are known, and the magnitude of the stresses in all except a@ and d. The 
polygon serves to determine these stresses. To find the stress in b we require 
to draw the polygon a, /,, 6a, /,, b, in which there are only two unknown 
stresses—those in 6a and b, the directions of which are however given. We 
cannot draw the polygon directly with the lines arranged in the manner shown 
in full lines, because 6a and 6 are not contiguous. If, however, the lines are drawn 
as dotted, we obtain a polygon which determines the stress both in 6a and in 
b, after which the lines may be rearranged in their natural order. By a similar 
process we find 68 and c, and can complete and check the drawing by adding 
the lines /’, 18 and /,. It is almost unnecessary to remark that la and 18 are 
equal, and that 6a and 68 are also equal. The sides of the polygon in fig. 35a, 
represent the loads on the elements of the machine in fig. 35, taken: in their 
natural order. The lines abcd in 35a represent the stresses on the links 
abed, or 1, 2, 3 and 4 of the machine in fig. 35. Let o be the point where 
a, b, c, and d intersect (fig. 35a), and let lines be drawn from o to the angles of the 
polygon ; now, draw the lines 2a and 2 in fig. 35, parallel to the lines of the 
same name in fig. 35a, and so placed that they abut against the ends of link a, 
and intersect in the line L,; draw 3a and 38, 4a and 48, 5a and 58 by a simi- 
lar rule. The octagonal frame thus obtained and shown by a broken dotted 
line is a frame which will be in equilibrium under the four given loads and 
the two stresses ine and 7 It is easy to see that this will be the case. The 
load L, and the two stresses in 2a 28 of the frame form a polygon in the 
reciprocal figure. The same relation obtains between the other loads and 
the links supporting them. Moreover, the links 2a and 58 in the reciprocal 
figure, give a closed polygon with the line 1a, representing the stressing. The 
lines abutting against the ends of falso give closed polygons with the stress 6, 
as shown in the reciprocal figure. The links 2a 28, &c., of the octagonal frame 
do therefore represent the directions of bearing pressures, at the joints on the 
hypothesis that there is no friction. The frame found by the method described 
will be called the loaded dynamic frame without friction. 
The elements e and / have been represented as without mass; if their 
weight and inertia are to be taken into account, their load is to be referred to 
the joints in a manner similar to that indicated for the other elements. Links 
1 and 6 would then be broken lines in the frame, with loads at their angles. 
It may be well to remind the reader that by hypothesis the machine is in equi- 
librium as a whole, and therefore in the reciprocal figure lines representing 
the loads necessarily form a closed polygon. 
§ 28. Loaded Dynamic Frame with Friction—The method by which we 
