193 



centre of strain, which is a close approximation when the pillar is very 

 long in proportion to its width. ^' 



The values of T or C in dilferent pillars are proportional to the 

 number of fibres subject to strain, that is to hd, and ^ is obviously pro- 

 portional to d; so that we have the numerator on the right side of the 

 equation proportional to hd^. Again, assuming that the deflection curve 

 is a parabola, jfrom which it can differ but slightly, f we have 



but so long as the strain per sectional unit in the extreme fibres, to 

 which their change of length is proportional, is constant, R will vary 

 in the same ratio as d ; and we have, therefore, h proportional to 



P 



"Whence, by substitution, 



^=ir*|, n. 



in which ^is a constant depending on the elasticity of the material, 

 which may be determined by experiment. 



If the pillar be round, and if d represent the diameter, 



ir=x|, III. 



which proves that the strength of long round pillars varies as the 4th 

 power of their diameter, divided by the square of the length ; and the 

 longer the pillar is in proportion to its diameter, the nearer will this 

 formula represent the truth. 



As all the longitudinal forces at the middle of the pillar balance, we 

 have the following equation : — 



which enables us to predict how a long pillar will fail, whether by the 

 convex side tearing asunder, or by the concave side crushing. A wrought 

 iron pillar, for instance, may be expected to fail on the concave side, as 

 its power to resist crushing is less than that to resist extension. A long 

 pillar of cast iron, on the contrary, will probably fail by the convex 

 side tearing asunder, as the compressive strength of cast iron greatly 

 exceeds its tenacity. Further, the eff'ective strength of wrought iron 

 to resist crushing is about 12 tons per square inch, while the tensile 

 strength of cast iron is nearly 7 tons per square inch ; and hence we 



* Mr. Hodgkinson's experiments show that this investigation is not applicable to 

 cast iron pillars whose length is less than about 30 times their width : even with such 

 short pillars it requires certain modifications, which he has deduced from experiment, 



f The curve will probably be intermediate between a parabola and a circle, approach- 

 ing the latter if the pillar taper towards the ends. 



