207 
mathematical theory of suspension bridges. 
forces be represented as in the ordinary curve by the incre- 
mental triangle P rp. But now x : y : : a. And by a 
repetition of the former steps x = ^==- . 
But on the principle of equal strength, 
As a : V a 2 + £ a : : z : £ 
therefore z = a x , — and 
V a 2 + 
Equ. D.) z—ay. nat. log. — ^ ’ 
i . . . £ z 
and by substituting a x ^7==- for 2; m the equation x= 
x = a x ; consequently, 
a -j- Q 
Equ. E.) — x nat. log.- - . 
Again, from the first analogy, y = , 
substitute for x its equal a x — 1 • , and 
y = a‘ x : therefore, 
Equ. F.) jy = the cir. arc. of which f is the tangent to radius a. 
a and y being given to find Multiply by 57 0 , 29578 
(the tab. log. 1.7581226) and reduce the decimals of a degree 
into minutes and seconds ; then will the tangent of that arc, 
multiplied by a, be equal to £. 
And when £ has been determined, the other columns of 
Tables III. and IV. are constructed from the above theorems, 
in a manner perfectly similar to that used in calculating of 
Tables I. and II. ; and they may be illustrated by the same 
example ; observing that a , now represents the uniform ten- 
sion on each given magnitude of iron throughout the chains, 
