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LXXVIII. On the Depression due to a Load at the Centre of 

 an Elastic Chain tightly stretched between Two Points in the 

 same Horizontal Plane. By Charles H. Lees, D.Sc* 



THE ense and convenience of determining Young's 

 Modulus for a material by stretching a wire between 

 two supports in the same horizontal plane, and measuring 

 the depression at its centre produced by a load suspended 

 there t, has led me to enquire more carefully into the theory 

 of the method, with a view to defining the limits within 

 which the method is applicable. In the theoretical work 

 known to me, the weight and flexural rigidity of the wire 

 are neglected, so that it may be taken as two straight lines 

 extending from the point at which the load is applied to the 

 two supports. 



When the weight of the wire is taken into account but its 

 flexural rigidity still neglected, the case becomes that of a 

 chain, the links of which increase in length under longitudinal 

 stress by stretching of their longitudinal parts without bending 

 of their transverse parts. The cmwe of equilibrium of the 

 chain on each side of the load is then a catenary, the lowest 

 point of which is beyond the point of application of the load. 

 When the load is withdrawn the whole chain becomes a 

 catenary with its central lowest point at a distance below its 

 ends which depends on the weight of the chain and the 

 stretching force applied. 



Let AB (fig. 1) be the two points of support at a distance 

 2/ apart, d the centre of the chain, M the mass suspended at 

 it, D the lowest point of the catenary of which BC 4 forms an 

 arc. Let the axis EO of the catenary be at a distance 

 c below 1), and let EO = ^. 



The equation to the catenary in terms of rectangular 

 coordinates with as origin is 



, e-\-x 

 y — c cosh . 



* Communicated by the Author. 



t Mr. G. F. StradliiiL r of Philadelphia has been good enough to furnish 

 me with some notes on the history of this method. It was first used by 

 s'Gravesande, Vhyxices Elcmenta Mathematical 1742, to determine the 

 breaking stress in wires : then by E. Tacke, Dissertation, Griefswald, 1889, 

 who considered it unreliable; and afterwards by Stradling, Ann. der 

 Phygtk, ifi. p. 330 (1890), ^vho found it both reliable and convenient. 

 Mr. Grime and I came to the same conclusion as the result of the experi- 

 ments we described in the February number of this Magazine. 



