150 ■ LECTURE XIII. 



ad^'antag•e to a ship's sailing; if therefore we sacrifice too much resilience to 

 strength, and too much of both to stiffness, we may perhaps create greater 

 evils than those which we wish to avoid. 



We have hitherto supposed the beams, of which the strength has been com- 

 pared, to be prismatic, that is, of equal breadth and thickness throughout, 

 which is not only the simplest form in theory, but the most generally useful in 

 practice. If however we have the power of giving any form that we please to 

 materials of a certain weight, which may often be done where several smaller 

 pieces are to be cut out of a larger one, or a larger one to be composed of 

 several smaller ones, or where the materials are either ductile or fusible, it is 

 frequently possible to determine a more advantageous form than that of an 

 equable beam or column. For since the extension which the parts of the 

 substance admit, without giving way, is the limit of their strength, if the 

 depth of a beam be everywhere equal, and the curvature unequal, the frac- 

 ture will first take place where the curvature is greatest, and the superfluous 

 strength of the other parts will be lost; so that, in order to have the greatest 

 strength that a given quantity of materials is capable of affording in a beam of 

 given length, the form must be such that the strength may be everywhere 

 equal, the tension of the surface being equal throughout; and the depth 

 must be as much smaller as the curvature is greater. It is also necessaiy to 

 consider whether the substance is likely to be crushed, and whether it is li- 

 able to be broken by detrusion, rather than by flexure. Sometimes the depth 

 of the beam may be limited, and sometimes its breadth; or it may be required 

 that the breadth and depth may be always equal or proportional to each other, 

 and the force may be either applied at one end of the beam, or it may be 

 equally divided throughout its length ; it may also principally depend on the 

 weight of the substance itself; and the strongest form will be different, accord- 

 ing to the different conditions of its application. In the most common cases, 

 the outline must be either triangular, or parabolic, as if the point of the tri- 

 angle were rounded off" ; but the curves required are sometimes of much more 

 difficult investigation. (Plate X. Fig. 128. . 147.) 



The strength of bodies is sometimes employed in resisting torsion, as in the 

 case of the axles of wheels and pinions, rudders of ships, and screws of all 



