324 APPLIED SCIENCE 



elements of the frame be the four lines a, b, c, d, shown as thick 

 black lines in Fig. 39, p. 323, and let the elements e and / be 

 joined to these at points intermediate between their extremi- 

 ties. Each element is then in equilibrium under the action of 

 three forces, and the simple dynamic frame is the quadrilateral 

 2, 3, 4, 5, having its angles on the prolongations XX and YY 

 of the directions of the stress in e and /, and having its sides so 

 placed as to pass through the joints ab, be, cd, da, denoted by the 

 letters MNPQ. It is not quite obvious how this quadrilateral 

 anay be drawn. It may be proved that all quadrilaterals of 

 which the angles lie in the lines XX and YY and of which 

 three sides pass through M, N, and P, have their fourth sides so 

 placed as to intersect at one point E ; the point E can therefore 

 be found by drawing two trial quadrilaterals, and this point 

 can then be joined with Q and so give the direction of one side 

 of the desired quadrilateral ABCD. Professor TAIT, who 

 pointed out this fact to the writer, also showed that the point 

 E might be more simply found as follows : Produce MN until 

 it intersects e prolonged in X, join X with P : similarly, produce 

 NP until it intersects / prolonged in Y, and join YM ; the point 

 E lies in the intersection of XP with YM ; the line QE gives 

 the direction and position of one side of the quadrilateral ABCD. 

 A second quadrilateral has been drawn on the figure for the 

 sake of illustrating the form which it assumes when the fourth 

 point is q, chosen outside the angle XEY. 



When, as in Fig. 40, the four members a, 5, e, and d, are 

 all loaded, the problem becomes still more complex. The octa- 

 gonal equilibrated polygon for the four loads and two stresses 

 in e and /, otherwise named 1 and 6, are shown in Fig. 40, with 

 lettering analogous to that employed for the simpler cases. 

 This polygon was formed in a somewhat indirect manner, and it 

 is probable that a simpler geometric method may be found if 

 the case should arise frequently in practice. (1) The relation 

 between a stress in e and one in / was found by a simple frame 

 and reciprocal figure, Fig. 41 . (2) The same process was repeated 

 for a stress in e, and a stress L a between the elements a and d, 

 Fig. 42. (3) The process was repeated for a stress L & between 

 I and c?, Fig. 43. (4) The process was repeated for a stress L c 

 between c and d, Fig. 44. (5) By addition the stress in e was 

 found which was required to overcome the given stresses due 



