328 



EDWARD SANG ON CASES OF 



that of the supporting members must be 2n, which may rest on 2n separate 

 points in the ground, but which may be brought together in pairs or otherwise. 

 If they be placed in pairs there are as many supporting as supported points. 



Robinson's octahedral stand shows this arrangement Avhen there are three 

 supported points ; we shall now take the case when four points are supported 

 from four points in the ground, as in fig. 11, where the connected points A, B, 

 C, D are shown as supported by the eight members AE, EB, BF, FC, CG, GD, 

 DH, HA. 



In general, that is when there is no regularity, such a structure contains all 

 the elements of stability. The positions of the foundation points being known, 

 if the lengths of the twelve members be prescribed, we shall have twelve equa- 

 tions of condition whereby to compute the twelve co-ordinates of the four points 

 A, B, C, D. Or, viewing the matter from the mechanical side, external pres- 

 sures applied at A, B, C, D may be resolved into their elements in three 

 assumed directions, x, y, z, and so may the stresses on the various parts ; there 



must be equilibrium at each of the points, 

 in each of the three directions, and so 

 again we have twelve equations whereby 

 to compute twelve unknown quantities. 



The algebraist at once perceives that 

 the resulting divisor (or determinant as 

 it is called) may happen to be zero, in 

 which case the stress becomes infinite ; 

 that the dividend may be zero, showing 

 that the particular member has no 

 strain upon it ; or even that both the 

 dividend and the divisor may be zero at 

 once, showing the structure to be inde- 

 terminate. But such investigations dis- 

 tract the attention from the objects under 

 consideration to their mere representative 

 symbols, and do not carry intellectual 

 conviction along with them. Their true 

 and highly important office is to deter- 

 mine accurately the various stresses, 

 thereby enabling the constructor to ap- 

 portion the strengths of the various parts. 

 The determinateness, that is the 

 stability, of a structure typified by fig. 11 

 ceases when we introduce symmetry or even semi-regularity. Let, for example, 

 the figures EFGH, ABCD be rhomboids, having their middle points and P 



Plan, 



Fig. 11. 



