2 Quantity of Iron necessary in a Tension Chain Bridge. [Jan. 



difficulty in the way of making every portion of iron in a complicated 

 structure exactly proportional to the tensions, and no portion must 

 be thinner, otherwise the loaded bridge would be in danger of falling, 

 and therefore the probabilities are that many parts would be thicker 

 than absolutely necessary. And therefore, as we have said, the economy 

 of iron will be practically greatest in bridges where the varieties of ten- 

 sion are least. This tells, then, in a practical point of view against the 

 Taper-Chain system in the question Taper-chain versus Common- chain 

 bridge. 



We shall now proceed to the demonstration of the property enun- 

 ciated, first, however, proving the following lemma which we shall find 

 of use in the course of our investigation. 



Suppose, in the first instance, that the bridge is as is represented in 

 fig. 1. This is given as a simple case to which we shall refer subse- 

 quently as a standard. The road-way is supported by two rods AB, 

 AB, proceeding from the piers, and attached to the road-way at B and 

 B. The tensions of these rods will not only support the weight of 

 the loaded road- way, but will produce a tension in the line BB, which 

 must be provided for by inserting a rod of iron, BB, of a proper 

 thickness, i. e. proportional to this horizontal tension, to prevent the 

 suspending rods from tearing the road to pieces. The rods AB, AB 

 must be held down by bolts, as shown in the diagram. Let C be the 

 middle point between B and B ; and Cb be drawn perpendicular to AB 

 produced. 



Lemma. — The quantity of iron in AB and BC necessary to resist 

 the strains is equal to a bar of the thickness at A, and of the length 

 Ab. 



Draw CB perpendicular to BC and meeting AB produced in D. 



The tension of BA at B is balanced by two forces, (1) the ten- 

 sion of BC, and (2) the portion of the weight sustained, acting in 

 BW. 



The triangle BCD has its sides parallel to the directions of these 

 forces, and these sides are therefore proportional in magnitude to the 

 three forces. 



