170 Analysis of Scientific Books and Memoirs. 



ture * proposed was not sufficient for insuring such a degree of strength and 

 permanence, as would be consistent with the due execution of a great na- 

 tional work. Wishing, however, to take upon himself the full responsi- 

 bility for the increased expence which such an alteration must necessarily 

 involve, Mr Gilbert printed, in the Quarterly Journal of Science, some 

 approximate investigations, and which confirmed him in the opinion he 

 had originally advanced as a member of the Commission. The result was, 

 that the interval between the points of support and the road- way was aug- 

 mented to fifty feet ; and the bridge now possesses, to adopt the language 

 of the distinguished author of the paper, " that full measure of strength 

 which experience has established as requisite and sufficient for works of 

 iron not perfectly at rest." 



Mr Gilbert has not only in the paper before us most fully investigated 

 all the necessary formula? for the ordinary catenary, and adapted tables to 

 them, but also added formula? and tables for the catenary of equal strength, 

 " a curve not merely of speculative curiosity," as he judiciously remarks, 

 " but of practical use, where a wide horizontal extent may chance to be 

 combined with natural facilities for obtaining a corresponding height for 

 the attachments." 

 By assuming the following elements ; viz. 



a — the tension at the apex, estimated in measures of the chain ; 



x = the absciss, the versed sine, or depth of curvature ; 



y = the ordinate, or semi-transverse length ; and 



z — the length of the curve ; 

 and considering that the tension a acts horizontally at the apex A, the weight 

 of the chain z perpendicularly, and the force of suspension in the direction 

 of the tangent to the curve ; these forces may be represented in direction 

 and magnitude by the elementary triangle Prp ; and hence we have 

 x : y : '• a : a, or x * '• y ''^ '• a > 



OTxt+^'.X-.'-^ + ^-.Z*, 



or since from the nature of the elementary triangle x 2 -f y 2 = % 2 , 

 we have 3*: 7 : : a 2 -fz 2 : * 2 > 



whence x =. 



•/(« 2 +* 2 ) 



and taking the fluent k = ^/(a 2 -f z *) — a, which is the first equation of 

 condition. 



Again, since x'y : : z : a, we have y aej a x • Substituting in this last 



* The depth of curvature here alluded to cannot be too particularly attended to 

 by engineers. In some plans of suspension bridges that have been proposed for 

 erection, the strength which a proper versed sine for the catenary affords, in propor- 

 tion to the horizontal interval between the points of suspension, appears to have 

 been entirely lost sight of. To diminish the expence as much as possible, the towers 

 abofc the road-way have been proposed much too low, thus sacrificing strength in 

 no inconsiderable degree. We beg earnestly to call the attention of engineers to 

 this important subject. 



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