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mathematical theory of suspension bridges. 
ordinate. If x be expressed in parts of the whole chain, 4 x 3 
will be the correction for the difference between the perifery 
and double ordinate. 
If x (the elevation at each end) be one link of the common 
measuring chain, 4 x 2 = of a link, f of of a foot 
= 0.3168 of an inch, varying as the squares of oc. 
If half the chain were considered as a straight line, and the 
hypothenuse of a right angled triangle, the horizontal dis- 
tance would be z — > giving but one half °f the true 
difference, 0,1584 parts of an inch. 
And if the chain were supposed to be in the arc of a circle, 
% =y -f- , &c. Andy = V zax—x z (when x is very 
small in comparison with a ) = V z a x . Therefore , 
And since y is also small in comparison of a , the se- 
cond term of the series will be the difference between 
the ordinate and the arc. Then substituting for a a , 
^ ; or if a: be expressed in parts of the whole chain, 
= j x* will be the whole correction, = 0.2112 parts of an 
inch, or two-thirds of the true difference. 
Formulas might readily be constructed for different eleva- 
tions of the extremities of the chain, but they would prove 
much too complicated for practical use. 
One further observation may be applicable to suspension 
bridges, wholly unconnected with the preceding investi- 
gations. 
In the event of their wanting stability to counteract and 
restrain undulatory motion, the ballustrades may be carried 
