172 Analysis of Scientific Books and Memoirs. 



This example is sufficient to explain the great practical utility of tlie 

 table. 



In the catenary of equal strength, Mr Gilbert introduces the symbol £ 

 as the representative of the mass of the chain. Then, since the forces 

 operating on this peculiar catenary, may be represented as in the ordinary 

 curve, by the same incremental triangle as before, we shall have 



*:#::£: a, 

 and which will produce by a repetition of the former steps, the fluxional 



nation i=_|£_. 



But on the principle of equal strength, 



Consequently * = h / f Sf ^ 



and which by taking the fluents, produces 



By substituting also the value of 2 just determined, in the equation 



x = if "j. 1 *t*V we shall have x = - — 4 . iu » 

 and which taking the fluents, produces 



• * == T na s 1 — ' 



From the first analogy also, we derive y = — — , 

 and which by substituting for x its value before derived, produces the 



equation y = T^TH* 



or by taking the fluents 



y = Circ. arc of which { is the tangent to the radius a, 



From these theorems Mr Gilbert constructs his third and fourth tables, 

 the application of which he also illustrates by an example. 



Assuming in the ordinary catenary, that x = 65.85 measures, is the 

 height of the attachment to give a maximum extent of span, with any vir- 

 tual tenacity of material, a will be 85 measures, and a + .r =85 + 65.85, 

 or 150.85 measures, equal the given virtual tenacity. This taken as before at 

 I of 4- of 14800 feet, will give 10875 feet for each measure, and the whole 

 span at 2y = 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 semispan being equal to the cir- 

 cular arc, of which £ is the tangent, to radius a, it is obvious that a X semi- 

 circular arc, must be the limit of the span. Therefore, if a = g of j, of 

 14,800 feet, or 1641.44, we shall have a X J 2 5154 feet. 



