OHAP. Vm.] CONTINUOUS GIRDERS. 133 



quantities, have deduced the general properties of the equili- 

 brium polygon, and will now endeavor by their aid to draw the 

 polygon itself. We shall then be able to find the actual mo- 

 ments A A' and B B' at the supports. It is impossible at first 

 to draw any single side of the polygon in true position, and we 

 must, therefore, endeavor to find certain points of the same suf- 

 ficient to determine it. 



Lay off first the three spans, PI. 14, Fig. 52 (d). Suppose 

 the second side of the polygon prolonged till it intersects the 

 vertical through the end support a, in a point K, Fig. 52 (J) 

 This point is known. It is given by the moment of the para- 

 bolic area in the first span with respect to this end support. 

 This moment we have already by the cross-lines in Fig. 52 (c). 

 We have then simply to take it off in the dividers from (o) and 

 lay it off from d to K' in Fig. 52 (d). We have now in Fig. 52 

 (b\ two points of the polygon known, namely, the end support 

 and K, which last must be in the second side prolonged. 



The triangle L M N is now of special importance. What- 

 ever may be the position of K M and M N, we have already 

 seen that the intersection M must always lie somewhere in the 

 limited third vertical. The first side KM must, however, 

 always pass through K, a known point. The second must pass 

 through the support, also a known point. The points L and N 

 must, moreover, always lie in the third verticals, distant from 



A, 5- \ and -^ 4 respectively. 



O O i , 



If the line K M takes up various positions under these con- 

 ditions, the line M N will revolve about a fixed point which is 

 given ~by the intersection of a line through K and the support 

 A with M N. 



If, then [Fig. 52 (d)~\, we draw a line in any arbitrary direction 

 through K', and note the intersections I/ and M' with the first 

 third vertical and the limited third, then through I/ and the 

 support draw a line to intersection N' with second third verti- 

 cal, and join M' N', and finally through K' and the support 

 draw a line intersecting this last in I, the point I thus deter- 

 mined is a fixed point, and remains the same for any position 

 ofK.' M'. It is therefore a point on the fourth side M N of 

 the polygon. For the triangle I/ M' N' may have any posi- 

 tion, yet so long as its angles lie in three parallel fixed lines, 

 and two of the sides pass through two fixed points, the other 



