of varying Elasticity. 269 



In the case of traction the only positive strain is -7- or B, 



which is the same for all the media of which the bar may 

 consist. If, then, the traction F be increased till B exceed 

 the limit of safety of an}?- one material, that material will 

 finally give way. Theoretically, of course, this material might 

 give way uniformly all round, with the result that the traction 

 F would then have to be supported by the remaining mate- 

 rials. This would lead to increased strain in all these ; but 

 the structure as a whole would still be safe if this new strain 

 were less than the limit of safety for each of the materials left. 

 If the increased strain exceeded this limit then a second rup- 

 ture would occur, and so on. In practice, owing to some want 

 of symmetry in the distribution of the traction, or to slight 

 inequality in the material, the first yielding material would 

 probably crack and give way only in the neighbourhood of 

 one point. This would alter the distribution of the traction, 

 and might bring it to bear most largely on the strongest 

 materials. If, however, the result were that the line of action 

 of the resultant of the tractional forces got displaced to a finite 

 distance from the axis of the cylinder, the strain would be 

 considerably lessened at some points and considerably increased 

 at others. It is obvious that such local increase of strain 

 would be extremely dangerous. Thus the traction to be ap- 

 plied with safety to a composite bar of this kind should be 

 calculated on the basis of the resultant strain not exceeding 

 the limit of safety of the material for which the limit is least. 

 In the case of longitudinal pressure B is negative, and for 

 a bar of one isotropic material the other two principal strains 



(viz. -r and -) are each equal to — <xB. In accordance with 



the theory recently referred to, — crB should not exceed the 



limit of safety as determined for the material by longitudinal 



traction. Others hold that the compression (i. e. B taken 



numerically) should not exceed a certain independent limit 



obtained from pressure experiments. The results we have 



obtained will enable the greatest pressure to be calculated 



which can be applied to a composite bar, without passing the 



limit of safety, for any one of the materials, determined in 



accordance with either of the above theories. 



In the case of torsion the only existent strain is the shear 



dv 



— =Er, which is shown in anv treatise on elastic solids to be 



equivalent to an equal extension and compression in directions 

 making each an angle of 45° with the direction of shear, the 



