322 The Ohio Journal of Science [Vol. XX, No. 8, 



put this in other words a spring in the form of a ruler laid fiat 

 wise would weigh exactly as much as one in the form of a ruler 

 laid edgewise provided only that in both cases the same load 

 produced the same deflection with the same fiber stress. At 

 first sight this seems almost absurd and to contradict our 

 experience, but a moment's consideration shows that the reason 

 of our doubt is that in applying the formulas for strength and 

 fiexture, the length of our beam is almost invariably given in 

 advance and we forget that a short flatwise ruler will be as stiff 

 as a long narrow one edgewise. 



In an exactly similar manner we may investigate the case of 

 a beam of constant strength and of constant depth, or one of 

 constant strength and constant breadth and we will find that 

 the expression for the weight of the beam is identical in both 

 cases. In the case of the hollow cylinder there are two limiting 

 cases, first, where the walls are very thin, and second, where 

 the cylinder became a solid rod. The follovv'ing table gives the 

 results. 



Beam of constant strength loaded at one end / ^E \ 



fixed at the other W= 3 ( ~p~ ) ^ ^ 



/ wE \ 



Hollow cylinder walls very thin W= 61 p 1 -t^ ^ 



Rectangular Beam W= 91 ^^ 1 r a 



Round rod W = 12 \~jr] ^ ^ 



It is to be noted that in the above expressions for the weight" 

 of the several types of beams that if we take the weight of the 

 beam of constant strength equal to one the weight of the others 

 will be 2, 3, and 4. Attention should be called to the great 

 superiority of the beam of constant strength when the depth 

 is made constant, the plan, in that case, being a triangle. In 

 this case not only is the total weight the least of the cases con- 

 sidered, but the weight is nearly all concentrated at the fixed 

 end, while the free end which moves is the vertex of the triangle 

 and consequently of almost no weight. 



We will now give the formulas for design for a few cases 

 which will sufficiently illustrate the method so that the designer 

 can readily apply the same process to any case that he might 



