ON PASSIVE STREKCTII ANT) FRICTION. 129 



bent in any degree that the nature of tlie substance will allow. If the co? 

 lumn was originally bent, any force, however small, applied to the extre- 

 mities of the axis, will increase the curvature according to the same law, but 

 if the column was originally straight, it cannot be kept in a state of flexure 

 by any longitudinal force acting precisely on the axis, unless it be greater 

 than a certain determinate force, which varies according to the dimensions of 

 the column. It is not however true, as some authors have asserted, that 

 every column pressed by such a force must necessarily be bent; its state 

 when it is straight, and submitted to the operation of such a force, will re- 

 semble a tottering equilibrium, in which a body may remain at rest until 

 some external cause disturbs it. The figure of a column naturally straight, 

 but bent a little by a longitudinal force, will coincide with that of the har- 

 monic curve, in which the curvature is as the distance from the basis. (Plate 

 IX. Fig. 117. . 121.) 



Considerable irregularities may be observed in all the experiments which 

 have been made on the flexure of columns and rafters exposed to longitudinal 

 foi ces ; and there is no doubt but that some of them were occasioned by the 

 difKculty of applying the force precisely at the extremities of the axis, and 

 others by the accidental inequalities of the substances, of which the fibres must 

 often have been in such directions as to constitute originally rather bent than 

 straight columns. ' 



When a rod, not very flexible, is fixed at one end in a horizontal position, 

 the curvature produced by its own weight is every where as the square of the 

 distance from the other end: and if a rod be simply supported at each end, 

 its curvature at any point will be proportional to the product of the two parts 

 into which that point divides it. But when the weights are supposed to be 

 applied to any given points of the rod only, the curvature always decreases 

 uniformly between these points and the points of support. (Plate IX. Fig. 

 122, 123.) 



The stiffness of any substance is measured by the force required to cause it 

 to recede through a given small space in the direction of the force. It is only 

 necessary to consider this property with regard to forces applied transversely; 

 In such cases the stiffness is directly as the breadth and the cube of the depth 



