322 



EDWARD SANG ON CASES OF 



abstract arrangement of the parts as represented by straight lines, there is the 

 problem, far more difficult, requiring much more constructive skill, of contriving 

 the manner of the junctions. 



In general, the relative positions of three points, as A, B, C, are determined 

 by the lengths of the three lines AB, BC, CA, joining them two and two. A 

 pressure applied at the point A can be resisted by the linear members AB, AC 

 only when its direction is in the same plane with them, and they must be 



enabled to offer resistance by pressures applied at 

 B and at C, which again, if they be not in the 

 direction BA, CA, must cause a stress on BC. 

 Hence we have here a system of six pressures in 

 equilibrium all having their directions in one plane. 

 Let a A and />B be the directions of the pressures 

 applied at A and at B, and let these be continued 

 to meet in ; join also CO. According to the 

 Fig. 1. known law of equilibrium, the strains on AB and 



on AC are proportional to the sines of the angles OAC and BAO, which again 



are represented by the doubles of the expressions and • where- 



fore, if we denote the strain on the member AB by the symbol AB, we have 



-ttt" • And on examining the equilibrium at B, 



we find also AB • -jjf- = CB • -™- , so that the direction of the pressure applied 



at C must also pass through the same point 0. 



Again, on comparing the pressure applied at A with the strain on AB, we 

 find 



ABC AOC 



n ,. — BOA _ 



the equality AB • AB =CA 



aA : AB 



CA.AB CA.AU 



whence 



— AOC _ — ABC 

 aA ' ~MT ~ Aii ' AB 



and it follows that the six expressions 



COB. BOA 



-j BOA .AOC 



aA ' AO 



bB 



BO 



6-C 



AOC .COB 

 CO 



^ ABC. BO A g^ ABC. COB 



AB 



BC 



are all of equal value. 



On multiplying each of the first three by 



AO.BO.CO 

 BOA.AOCCOB 



CA- 



ABC .AOC 

 CA 



