332 



EDWARD SANG ON CASES OF 



Here, in order to support five points, sixteen members are conjoined, and 

 yet there does not seem to be any redundancy ; moreover, the arrangement is 

 quite symmetric. The equilibrium at each point gives rise to three equations 

 of condition, and these fifteen equations cannot possibly serve to determine 

 sixteen strains. But if we apply a pressure at any one of the five points, the 

 fabric resists it, the various members are strained somehow, the law of equa- 

 tions notwithstanding. The explanation of this paradox may afford an instruc- 

 tive exercise to the student. 



Fig. 17. 



When the five points are arranged in the corners of a pentagon, each being 

 carried by two supports, as shown in plan by fig. 17, the structure is rigid, pro- 

 vided the polygons be convex. Of this we easily 

 convince ourselves by supposing one of the con- 

 nections, say EA, to be removed, and by ex- 

 amining the motion of the link system thus left. 

 The point A can move only in a circle, having 

 KF for its axis ; let A be moved inwards, the 

 member AB will then cause the triangle FBG 

 to turn outwards on FG as a hinge ; BC will 

 draw C inwards, CD will push D outwards, and 

 lastly, DE will draw E inwards ; wherefore the 

 distance AE will be shortened, and the member AE can be replaced only when 

 the structure is brought back to its former position. 



Following this line of argument one step further, we see that in the case of 

 a hexagon the first and last points would move, the one outwards, the other 

 inwards, and that so the distance might remain unchanged. When the hexa- 

 gons are semi-regular or halvable, the distance remains absolutely unchanged, 



and the structure is indifferent as to position. This 

 same remark applies to all polygons of an even number 

 of sides. 



If, however, the upper and lower polygons be placed 

 conformably, as in fig. 18, the structure is rigid, whether 

 the number of supported points be even or odd. 



Fig. 18. 



These truths may be illustrated experimentally by 

 preparing a few isosceles triangles as AFB, having 

 perforations at A and B, through which an elastic 

 string may be passed. On connecting a number of these, say seven, by a con- 

 tinuous thread, and spreading them out on a table, we form a flexible equal- 

 sided heptagon, and when this is arranged regularly the points F are in the 



