140 lECTURE XIII. 



of the beam, and Inversely as the cube of its length. Thus if we have & 

 beam which is twice as long as another, we must make it, in order to ob- 

 tain an equal stiffness, either twice as deep, or eight times as broad. The 

 property of stiffness is fully as useful in many works of art as the ultimate 

 strength with which a body resists fracture: thus for a shelf, a lintel, or a 

 chimney piece, a great degree of flexure would be almost as inconvenient as 

 a rupture of the substance. 



When a beam is supported at both ends, its stiffness is twice as great as 

 that of a beam of half the length firndy fixed at one end ; and if both ends 

 are firmly fixed, the stiffness is again quadrupled. For if the whole beam 

 were inverted and supported by a fulcrum in the middle, each half would re- 

 semble a separate beam fixed at one end, and the fulcrum would bear the 

 sum of two equal weights placed at the extremities, disregarding that of the 

 beam ; and consequently the same flexure will be produced by placing a 

 double weight on the middle of the beam in an inverted position. If both 

 ends were firmly fixed, the curvature would be every where as the distance 

 from the middle of each half, the whole being in the same state as four se- 

 parate beams fixed at their extremitii^s : each of these beams would be eight 

 times as stiff as beams of -twice the length, and the whole beam, in this state, 

 would be eight times as stiff' as if the ends were simply supported. It is, 

 however, difficult to fix the ends of a beam so firmly as to increase its resistance 

 in this proportion, unless it be continued both ways considerably beyond the 

 supports. 



It is evident that a tube, or hollow beam, of any kind, must be much 

 stiffer than the same quantity of matter in a solid form: the stiffness is indeed 

 increased nearly in proportion to the square of the diameter, since the cohe- 

 sion and repulsion are equally exerted with a smaller curvature, and act also 

 on a longer lever. 



Torsion, or twisting, consists in the lateral displacement, or detrnsion, of 

 the opposite parts of a solid, in opposite directions, the central particles only 

 remaining in their natural state. We might consider a wire as composed of 

 a great number of minute threads, extending through its length, and closely 

 connected together; if we twisted such a wire, the external threads would 



