268 Mr. C. Chree on Bars and Wires 



first a succession of materials in contact, we see from (29 a) 

 and (30 a) that, supposing the end z=0 fixed, a possible 

 solution for any medium is 



v=Erz (37) 



Further, this solution, regarding E as an absolute constant, 

 will apply to all the media. For from (9 a) we see that the 

 only stress existing will be S, and at all the surfaces of se- 

 paration, as well as at the bounding surfaces of the cylinder, 

 fj, = v = ; thus all the surface-equations (10 a) are identically 

 satisfied. Also the value of v is the same for any two adja- 

 cent media at their common surface. If Gr be the couple of 

 torsion applied at the end of the cylinder, and if a 1 . . . a i} 

 ft, % . . . ftj have their previous meaning, we get, to determine E, 



G = 27rE|l W^r+l 2 n 1 r 3 dr + ...+ \ ftr 3 <ir|, . (38) 



L e. 



G= ™ [n(a l *-b*) + n l {af--at) + . . , + n^-a *)]. (39) 



The values of a and b and the number of the media may be 

 any whatever. 



With limitations similar to the case of longitudinal traction 

 this solution may be supposed to apply to a continuously 

 varying medium, and the value of E will then be given by 



G = 2tteP 



nr' 6 dr (40) 



This last expression obviously includes (38) , treating ft as a 

 discontinuous function. 



The chief use of the preceding investigations would pro- 

 bably be in assigning the limit to the traction, pressure, or 

 torsion which could be applied with safety to a structure of 

 the kind considered. Unfortunately there seems no general 

 agreement among practical men as to how the limits of safety 

 may be fixed for a material when exposed to any system of 

 force, except perhaps longitudinal traction. One theory that 

 seems to meet with considerable approval is that, whatever 

 the system of forces may be, the structure is safe so long as 

 the greatest positive strain does not exceed a certain limit, to 

 be determined experimentally for each separate material, pre- 

 sumably by uniform traction. As to the correctness of this 

 limit for the case of uniform traction no doubt need be 

 entertained. It does not follow that rupture will ensue as 

 soon as this limit is passed ; but the nature of the material 

 itself will be altered, and rupture will follow sooner or later. 



