142 CONTENTIOUS GIRDER. [CHAP. DL 



But we have also 



UoU^VoVilBUpiBVo ::Zj:^; 

 hence 



L U : L W : : I V x^ : I W x^ 



The ratio of the parts into which W U is divided by the 

 point L depends, therefore, for given spans only upon the ratio 

 of I V to I W , or upon the position of I alone. 



If, therefore, we were to draw through the point I another 

 polygon, the point L would be unchanged, or still more gener- 

 ally, if the point I for different heights of the supports and 

 different polygons moves in the same vertical, the point L will 

 also move in a vertical. 



If the supports are in the same straight line, the points I and 

 Li are the points of inflection of the elastic line. We have 

 therefore the principle, that if for different polygons the in- 

 flection point I remains the same, the inflection point L re- 

 maim also the same. 



The point I being given, we can easily construct the point L. 

 We have only to draw through I at will any line intersecting 

 the third vertical through V and the limited third at, say, V 

 and W. Through V and the support B draw a line to inter- 

 section Ui with third vertical through U . Join now I^ with 

 W. The line U t W cuts the line through the supports A and 

 B in the point L. (See also Art. 86, Fig. 52, d.) 



93. The '* Fixed Point." Suppose that, starting from the 

 left support A [PI. 15, Fig. 57], we have a number of unloaded 

 spans. The end A then is an inflection point, since the mo- 

 ment there is zero. Starting from this point, therefore, we can 

 construct, according to the preceding Art., the inflection point la 

 for the next span. Then starting from this we may construct 

 the point Is for the third span, and so on. Since these points, 

 under the assumption that the supports att lie in the same 

 straight line, do not change their position, whatever may be the 

 loading of the loaded spans, and whatever spans be loaded, we 

 call them faced points. 



A second Aeries of fixed points may be in similar manner 

 constructed, when a number of spans from the right are un- 

 loaded, so that there are two series of fixed points. In the 

 end spans the end supports are fixed points. 



It follows directly from the construction that the fitted points 

 are always within the outer third of the span. 



