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mathematical theory of suspension bridges. 
Assuming in the ordinary catenary that a? = 65.85 mea- 
sures, is the height of the attachment to give a maximum 
extent of span with any virtual tenacity of material, a will be 
85 measures, and a -f- x = 85 -f 65.85, or 150,85 measures 
equal the given virtual tenacity. This taken as before at 
~ of — of 14800 feet, will give 10,875 feet for each mea- 
sure, and the whole span at zy = 2175 feet. Chains merely 
supporting themselves, and at the utmost of their tenacity 
will extend nine times further, or to 19575 feet. 
In the catenary of equal strength, the semi-span being 
equal to the circular arc of which £ is the tangent to radius a, 
it is obvious that a x semi-cir. arc must be the limit of the 
span. Therefore if a — ~ of -k of 14800 feet, or 1644,44 
<ix- = 5154 feet. 
2 
And if the chains merely sustain themselves at their utmost 
tenacity, 5154 *9 will give 46385 feet, equal to 8,785 miles, 
or somewhat more than 8 miles and three-quarters. 
But this case is purely hypothetical, for the purpose of 
ascertaining a limit, since £, the mass or weight of the chain 
must be infinite, and consequently its length : the figure 
approaching indefinitely near to that of a chain sustaining 
itself from an infinite height, which figure is identical with 
that of a building, capable so far as pressure and the strength 
of materials are alone concerned, of being carried to any 
elevation whatever. This figure is readily determined : 
Let a = the section of such a building at its base, 
y == the section at any height, 
x = that height ; 
