MEMOIR OF EATON HODGKINSON. 211 



in my investigation, by my having assumed tlio momentum of the forces on each 

 side of the neutral axis equal to each other, instead of the forces themselves. 

 This paper did not come to my knowledge till the third edition of my essay was 

 neaily jtrinted off, and the correction could not then be made.'' The liev. Dr. 

 "Whewell, in his '' Anal^'tical Statics," refers to this paper, and gives the inves- 

 tigation of the neutral line on the same principle as that adopted by Mr. Ilodg- 

 kinson, who always maintained that "we could see no cause why it should 1)0 

 rejected, especially since it seems to us to be every wliere consistent and just." 

 (See ]\Ianchester Philosophical Society's Memoirs, vol. iv, p. 241.) It aj)pears 

 that his friend. Dr. Dalton, took great interest in the deductions of this paper, 

 and discussed them freely with huu as he proceeded with his experiments, which 

 \\ ill ever be regarded as marking an epoch in the subject of the strength oi mute- 

 rials. Indeed the theoretical investigations of this paper, though new and 

 important, form only a small portion of its merits. The experiments record(;d 

 in it established the laws, '' that the extensions of the fibres of a bent beam were 

 proportional to the forces during the early stages of flexure ; but as the exten- 

 sions arrived nearer to fracture they increased faster than the forces," and, '' that 

 so long as the forces are moderate and are applied in the direction of the fibres, 

 the compressions will be as the forces ; but when the beam becomes bent the 

 fibres, being then crushed, ofter a feeble resistance to the force." These results 

 W'ere obtained direct from the unerring voice of nature. The first of these laws 

 was announced by Dr, Robison, as a general law of nature, on the simple 

 authority of a few experiments on the slips of gum caoutchouc and the juice of 

 the berries of the white bryony, of which a single grain will draw to a thread 

 two feet long, and again return to a perfectly round sphere. (See " Manchester 

 Memoirs," vol. iv, p. 252.) 



'• On the Forms of the Catenary in Suspension Bridges," (read Februarv 8, 

 1828, vol. V.) 



The chain bridge at Broughton, Manchester, which broke down by a troop of 

 soldiers marching over it, and the celebrated Menai suspension bridge, built by 

 Telford, had stimulated inquiries respecting the best form of such structures. 

 These inquiries, naturally- enough, led to a reconsideration of the catenary , a curve 

 the properties of which, under given conditions, were first discovered by James 

 Bernoulli. (See Leslie's '' Geometry of Curved Lines.") 



In this paper a great degree of generality is given to the catenarian curve. 

 After the known j)roperties of the common catenary are clearly investigated, 

 the fonnulic are then applied with great ability to determine the form of susj)en- 

 sion bridges when the iceight of catenarian chain , the weight of the roadwai/, and 

 the ivcigJit of tlte suspension rods axa taken into account. The introduction of 

 these complex, though necessary, elements into the (piestion led to the forma- 

 tion of the following difficult and comprehensive differential equation: 

 adx 

 'dy 



where ar, y are the cuiTcnt co-ordinates of a point in the curve, and z the length 

 of the curve from this point to the lowest point. The explanation of the con- 

 stants a, &, c, e is as follows : 



a=the tension of the curve at its lowest point. 



?;=thc weight of a unit of length of the curve. 



c=thc weight of a unit of length in the roadway, which is supposed to bo 

 divjded transversely into separate parts, and may include any weight uni- 

 formly distributed over it, with that of the suspension rods below the 

 horizontal line. 



C=the weight of a imit of vertical surface in the suspeiuling rods, the rods 

 being here supposed to be uniformly distributed, and indefinitely near to 

 one another, atul therefore reckoned as a uniform surface. 



r 



'-=.lz-\-cy^c\xdg .... (A,) 



