134 CONTINUOUS GIRDERS. [CHAP. VIII. 



side must also pass through a fixed point.* Out of all possible 

 positions of the triangle, one of these positions must coincide 

 with the polygon sides, and hence this fixed point is a third 

 known point, since we have already K' and the end support. 



Although, then, we are as yet unable to draw any of the sides 

 of the polygon in true position and direction, still from the 

 hitherto known properties we have deduced a new one. We 

 know now a point through which the fourth side must pass. 

 But this is not all. We proceed still further. The fourth and 

 fifth sides must intersect upon the vertical through the centre 

 of the second span. These sides, moreover, cut off upon any 

 vertical the moment of the parabolic area with respect to any 

 point in that vertical. We know this moment thus for the 

 point I just found. It is found by taking the segment cut off, 

 from the vertical through I, by the cross-lines for the parabolic 

 areas found above in Fig. 52 (c). 



Laying this segment off from I, we thus find I', a point in the 

 fifth side prolonged. From this point we proceed as before to 

 find the next fixed point I". We then lay off from I" the mo- 

 ment of the parabolic area for this point and find I'", a point 

 upon the eighth side. We can now draw the polygon itself. 



Thus the eighth side passes, of course, through the last sup- 

 port and also I"'. It is therefore determined. Through the 

 intersection of this line with the vertical through the middle of 

 the span and the point I" the seventh side passes. The sev- 

 enth side is therefore determined. Through the intersection of 

 this with the third vertical and the support the sixth side 

 passes and continues till it intersects the third vertical on the 

 other side. Then from this point towards I' to intersection 

 with vertical through centre of middle span. From this last 

 point towards I to intersection with third vertical. From this 

 last point again through support to intersection with third verti- 

 cal on other side ; then towards K' to intersection with vertical 

 through centre of end span ; and lastly, from this last point of 

 intersection through the end support, and the polygon is com- 

 plete as given in Fig. 52 (b). 



In this manner, however, inaccuracies may occur. To avoid 

 these we may start from the right end support and also find 

 four fixed points as above. It is unnecessary to make the con- 



* This proposition the reader can easily prove geometrically or analytically. 

 See Art 112. 



