324 



EDWARD SANG ON CASES OF 



pressures at the corners of a tetrahedron balancing each other, the directions 

 must all tend to a single point. But this inference does not hold good ; it may 

 be that no two of these directions meet at all. 



At each of the four points we have the equilibrium of four pressures, 

 namely, the external pressure and the strains on the members meeting there. 

 These strains can be computed when the direction and intensity of the applied 

 pressure are known. 



Thus let us continue the direction dD of a pressure applied at D until it 

 meet the plane of ABC in some point O, and let AO, BO, CO be drawn. We 

 have then the equalities 



rfD JDA DB DC 

 DO . ABC ~ DA . BOC ~ DB . CO A ~ DC . AOB ' 



The points A, B, C and O remaining as they are, if D were brought nearer 

 to 0, the first of the above expressions would be augmented in inverse propor- 

 tion to DO, and if D were brought actually to 0, this term would become 

 infinite, the strains DA, DB, DC also infinite and the structure impossible. 



When, in such an arrangement as fig. 3, the resistances at A, B, C are in a 



direction parallel to dD, their intensities are proportional to the opposite 



triangles, so that 



dD_ ok 6B cC 

 ABC _ BOC _ COA _ AOB ' 



and thus the distribution of the pressure among the ultimate resistances is 

 independent of the distance DO. 



In fig. 3 the point is placed inside of the triangle ABC, and a pressure 

 <i applied in the direction (TDO causes compression 

 in all the three members, DA, DB, DC. In fig. 4 

 is placed outside of the line AC, and, with 



B 



Pie. 4. 



Fig. 5. 



Tie. 6. 



pressure in the direction e/DO, the members DA, DC are compressed, while 

 DB is distended. 



If here the point D were brought down to 0, the structure would take 

 the form of a plane tetragon ABCD, with its two diagonals AC and BD, as 

 shown in fig. 5. Such a structure can offer no resistance to pressures inclined 

 to its plane. 



If the point D were on the straight line AC, as in fig. 6, it might seem that 



