54: MOMENT OF ROTATION PARALLEL FORCES. [CHAP. V. 



sition for the line closing the polygon and new reactions. As 

 the span continues to shift to the right, the lines closing the 

 polygon revolve, and as their projections are always constant, 

 viz., equal to the span, they are all tangent to a. parabola, which 

 they therefore envelop. 



48. Propertie of this Parabola. This parabola has sev- 

 eral important properties which will aid us in the investigation 

 of the case above proposed.* In PL 9, Fig. 32 (d), let XX be 

 the line along which the span is shifted ; a M and a N the 

 outer sides of the polygon, intersecting at a, along which the 

 closing lines slide as they revolve. For a given position 8 s of 

 the span, IT a- is the corresponding line. s is the position of 

 the span, for which the centre, C , lies in the vertical through a. 

 In this position <T O o- is tangent to the parabola at o> , its middle 

 point, and upon this line lie the centres of all the other lines 

 (taken of course as reaching from a N to a M). Now the 

 point of tangency, /3, of any other line, as <r <r, with the parabola, 

 is as far from the centre of that line, 7, as the centre of that 

 line is itself from c . We have then only to make c b equal to 

 c c , and drop a perpendicular through b to find /3. Thus for 

 the position s^ Si and the line <r- L <r l5 to find the point of tangency 

 !, make <J X d^ equal to c t C , and draw d t Si perpendicular to 

 intersection with cr^ ov 



Inversely we may find that position for the span s s, for which 

 the vertical through a given point, b, shall pass through the 

 point of tangency. 



We have only to move the span so that its middle point c 

 shall be as far from c as it is already from the given point, or 

 make c CQ equal to c b. (See Art. 75.) 



If we shift now the span s s, and at the same time the point 

 o through an equal distance, the intersections of the vertical 

 through b, with the corresponding closing lines of the polygon, 

 will all lie upon the same line tr or. 



If therefore b\ is such an intersection, b has been moved from 

 b to b\, and hence the span from s s to s^. 



49. Different Cases to be Investigated. We are now 

 ready to investigate the effect of a live load such as represented 

 in PI. 9, Fig. 32 (a). For the determination of the proportions 

 of the truss the following points are specially important : 



* See Elemente der Gra/phischen Statik, Bauschinger, pp. 108-114. Also, 

 Die Qraj)hisc7ie Statik, Culmann, pp. 136-141. 



