Mr D. Gilbert's Theory of' Suspension Bridges. 171 



found equation, the value of a? before deduced, 

 we have , = -^t- y 



and taking the fluent y = a X nat. log. — , which is the se- 

 cond equation of condition. 



By assuming also N for the number of which - is the natural loga- 

 rithm, supposing a and y to be given, and afterwards adopting M as the re- 

 presentative of a N — a, Mr Gilbert deduces the equation 

 M 2 

 * - 2M+2a' 

 which determines the value of the abscissa of the catenary. And from the 

 first equation of condition he obtains 



which determines the length, and consequently the weight of the chain. 



The tension also at the point of suspension P being equal to ij{a 2j {-z^), 

 becomes in the present case equivalent to a+x, and which furnishes, since 

 the value of x is known, the absolute value of the tension at P. 



From the analogy also which exists between the parts of the elementary 

 triangle, and of the forces corresponding with it, the angle of suspension 

 becomes known. 



To render the subject accessible to practical men, Mr Gilbert has, as be- 

 fore mentioned, constructed, from the preceding theorems, his first and se- 

 cond tables, and the application of which he has explained by the follow- 

 ing example. 



Suppose the span of a suspension bridge to be 800 feet, and the adjunct- 

 weight of suspension rods, road-way, &c. one-half the weight of the 

 chains. Then if the full tenacity of iron be represented by the modulus 

 of 14,800 feet, the virtual modulus for the whole weight must be reduced 

 in the proportion of 3 to 2, or to 9867. Let it also be determined to load 

 the chains at the point of their greatest strain, which is at the point of sus- 

 pension, with one sixth-part of the weight they are theoretically capable 

 of sustaining. 



Then since the semi-span is by the hypothesis 400 feet, and that y in 

 the first of Mr Gilbert's tables is taken at 100 measures, each of these 

 measures must in the present instance be 4 feet, and the weight expressed 

 in the same measures to be sustained at the points of suspension ; or, in 

 other words, the value of T, must be 9867-f-6 X 4, or 411.125. Entering, 

 therefore, with this value of T in the proper column of the table, we ob- 

 tain, with the greatest ease, the following values ; viz. 

 a = 400 measures or 1600 feet. 

 x = 12.563 measures or 50.26 feet. 

 z = 101.045 measures or 404.18 feet, 

 and the angle of suspension 75° 49'. 



