on the Flexure of Thin Bars. 323 



the amount of the allowance, in any case, is too small to have 

 led to the mistake with which I have immediate concern. 

 The mere effect of curvature, then, being supposed to be eli-' 

 minated, there will remain the effects of compression or dis- 

 tension. 



Taking a very crude view of the state of a bent bar, we may 

 assume, that lines drawn originally perpendicular to its length 

 when straight, remain normals to its surface when bent. 

 Crude though this view may be, it affords a sufficient approxi- 

 mation for my present purpose. We may then regard the two 

 dots on which the observations are made, as supported on two 

 pillars resting on the locus of neutrality ; and the distance be- 

 tween those dots (the reduction of the arc to its chord having 

 been allowed for) will be changed by a quantity, the product 

 of the height of the pillars by the sine of the inclination of 

 each end of the bent bar. 



Now, when the bar was supported in the middle, and its 

 ends allowed to hang down, the lengthening was found to be 

 scarcely half of the shortening obtained by supporting the 

 ends and allowing the middle to be depressed, the versed sine 

 of the arc being the same in both cases. Therefore, concludes 

 Captain Kater, the height of the pillars in the one case must 

 be scarcely half of the height of the pillars in the other : that 

 is to say, the neutral plane must be twice as far from the con- 

 cave as from the convex surface. 



It must, however, be clear to any one who bestows a single 

 thought upon the subject, that, since the half shortening or 

 lengthening is the product of the distance of the neutral plane 

 by the sine of the inclination, the comparison of the shorten -■ 

 ing with the lengthening cannot exhibit the relative dis- 

 tances of the neutral planes in the two cases, unless the incli- 

 nations of the curve be alike. Captain Kater has reasoned as 

 if the curve in the one case were a copy of the curve in the 

 other ; but unfortunately for his logic, the two forms of the 

 curve are essentially different, as may be known from the peru- 

 sal of the works of any respectable writer on flexure. When 

 a bar whose half length is /, is supported by its middle, the 

 depression of the end is \l\ multiplied by a certain constant de- 

 pending on the nature of the material and thickness of the bar, 



