114 LECTURE XIII. 



the number of the resisting particles, but also gives each of them a double 

 power, by increasing the length of the levers on which they act. The 

 increase of the length of a beam must also obviously weaken it, by giving 

 a mechanical advantage to the power which tends to break it ; and some 

 experiments appear to show that the strength is diminished in a proportion 

 somewhat greater than that in which the length is increased. 



The strength of a beam supported at both ends, like its stiffness, is twice 

 as great as that of a single beam of half the length, which is fixed at one 

 end ; and the strength of the whole beam is again doubled if both the ends 

 are firmly fixed. 



The resilience of a prismatic beam, resisting a transverse impulse, follows 

 a law very different from that which determines its strength, for it is 

 simply proportional to the bulk or weight of the beam, whether it be 

 shorter or longer, narrower or wider, shallower or deeper, solid or hollow. 

 Thus a beam ten feet long will support but half as great a pressure, with- 

 out breaking, as a beam of the same breadth and depth, w T hich is only five 

 feet in length ; but it will bear the impulse of a double weight striking 

 against it with a given velocity, and will require that a given body should 

 fall from a double height in order to break it. 



It is therefore of great consequence in the determination of the form and 

 quantity of the materials to be employed for any mechanical purpose, that 

 we should consider the nature as well as the magnitude of the forces which 

 are to be resisted. Stiffness, strength, or resilience, may be separately or 

 jointly required in various degrees. For a ceiling, stiffness would be prin- 

 cipally desirable ; for a door, strength ; for the floor of a ball room, resi- 

 lience ; for a coach spring, resilience and flexibility, that is, resilience with- 

 out stiffness. An observatory should be as stiff as possible, a ship as strong 

 as possible, a cable as resilient as possible. 



It is a common remark, that a floor which shakes is the strongest ; and, 

 improbable as it appears at first sight, it may perhaps be founded in truth ; 

 for if the absolute strength of a stiff and a shaking floor were equal, the 

 shaking floor would bear the effects of motion with the least injury. It is 

 possible that a stiff floor, which would support a numerous assembly, 

 might give way at a ball ; while a more resilient one, which would be 

 suited for dancing, might be destroyed by a crowded concert. 



A coach spring, divided into plates, has the same power of resisting, 

 without being broken, the momentum of the carriage, arising from sudden 

 elevations and depressions, as it would possess if it formed one entire 

 mass, while its greater flexibility allows it to regulate these motions in a 

 much more gradual and gentle manner. A single piece of timber may 

 perhaps, sometimes, have too much of the flexibility of a coach spring, its 

 strata sliding, in some degree, on each other ; in such a case its stiffness 

 and strength may be increased by binding it very firmly with hoops. 



The transverse strength of a perfectly elastic substance, fixed at one end, 

 is to its direct cohesive strength as the depth of the substance to six times 

 its length. This proportion is equally applicable to such substances as re- 

 sist compression more strongly than extension ; for their immediate repul- 

 sive force is probably not greater than their cohesive force, when their 



