INSTABILITY IN OPEN STRUCTURES. 333 



corners of a larger regular heptagon. If we now trace on a flat board a regular 

 seven-sided figure, intermediate in size between these two, and secure the 

 points F of the triangles in holes made at the corners, we shall have erected a 

 structure analogous to that shown in fig. 17, and shall find it to be rigid. 



If one of the triangles be removed, and the same process of construction 

 followed with the remaining six, the resulting regular hexagonal structure is 

 found to be instable. 



Another reduction of the number brings us to the pentagonal structure, 

 which again is stable ; and still another removal gives the tetragonal instable 

 fabric ; and, lastly, when only three triangles are left, we have Robinson's 

 octahedral stand. 



The important distinction between the two cases of conformable or of un- 

 conformable polygons may be illustrated by preparing two pairs of triangles, 

 one pair as EAB, GCD, of fig. 12, the other pair as FBC, HDA, and by con- 

 necting the sides, AB, BC, CD, DA, so as to form a flexible tetragon. 



When the feet, E, F, G, H, are secured in the corners of a rhomboid or of a 

 rectangle, the structure is rigid, if AB, BC be parallel to EF, FG ; in all other 

 cases it is instable. 



These cases of instability in open structures have been elicited by means of 

 the simplest considerations in Geometry and Statics ; they lie indeed on the 

 very surface of mechanical inquiry. They do not occur as isolated examples — 

 they are arranged in extensive groups ; and, being found in those classes of 

 structures which may be called shapely, they stand out as warning beacons to 

 those engaged in engineering pursuits. 



VOL. XXXIII. PART II. 3 D 



