6oo 



NATURE 



{Oct. 19, 1882 



1. Suppose the stone or iron piers to be much lower 

 than in the plans hitherto proposed, and suppose that the 

 top of a pier carries a bracket on each side, so that the 

 great suspending chain passes over the points of the 

 brackets, and its suspending action begins at those points. 

 The bracket frame may be horizontal where it passes the 

 top of the pier ; or it may be raised in a horn on each 

 side, and thus adapted to a smaller height of pier. By 

 this construction, with brackets 150 feet long(a trifle com- 

 pared with those of the proposed cantilevers), the piers 

 may without difficulty be shortened 200 feet, and the 

 acting-length of suspending chain may be reduced 150 

 feet at each end, or 300 feet over each water-channel. 

 This would leave much liberty in regard to the curvature 

 of the chain. 



2. It is very desirable, if possible, to reduce the specific 

 weight of the chains per yard, corresponding to a specified 

 suspension strain. This has been attempted on the Con- 

 tinent by the use of wire, and it has been highly praised 

 for its combination of lightness and strength. The 

 longest carriage- bridge that I have passed (that of Frey- 



. burg, S90 feet span) is a wire bridge. I have also crossed 



the Rhone at Mo. telimart by wire arches of considerable 



span. I know not whether this construction has been 



tried in England. G. B. Airy 



The White House, Greenwich, September 26, 1882 



Appendix 



Having adverted above to the measurement of the end- 

 wise or ''buckling" force upon a bar, I will here give a 

 theory, by application of which the admissible amount of 

 end-pressure in any case may be ascertained. 



The curvature of any point of a bar depends upon the 

 action of two causes. The first cause is the exernal force, 

 whose angular momentum or effect to bend the bar at any 

 point under consideration is proportional to the produ :t 

 of the force (expressed in multiples of a definite unit — as 

 the pound avoirdupois, or the ton, &c.) by the distance of 

 its line of action from the point under consideration 

 (expressed in multiples of the inch, or the foot, &c). The 

 second cause is the internal elastic force of the bar pro- 

 duced by curvature, whose tendency is to oppose the 

 bending action of the external force ; I shall assume the 

 magnitude of this force to be proportional to the curva- 

 ture, or inversely proportional to the radius of curvature, 

 at the point under consideration, its coefficient being for 

 the present expressed only as a symbol. The effects of 

 these two causes balance in a quiescent position of the 

 bar, and they must therefore be made algebraically 

 equal. 



The course of investigation will now be as follows : - 

 First, I shall give the"equation between force and curva- 

 ture when a bar is bent by a transversal force, acting at 

 the middle of its length. Second, I shall give the similar 

 equation when a bar, at least slightly bent, is exposed to 

 an end-wise force. ^The condition "slightly bent" is 

 necessary to exclude the absurdity of a very heavy weight 

 supported end-ways by a very thin wire.} In both cases 

 the results will contain the symbolical coefficient to which 

 I have lately alluded. From the first investigation I shall 



deduce the value of that coefficient. I shall substitute it 

 in the result of the second investigation ; and finally, 

 shall obtain a most convenient expression for the largest 

 admissible force acting endwise on the bar. 



(First). Theory of a bar supported at its ends and bent 

 horizontally by a force applied to the middle of its length. 

 The symbols are sufficiently explained in the diagram. 

 It is indifferent, practically, whether the support of either 

 end of the bar against the force w bi a pin (as on the left 



side), or a force - (as on the right side) ; the latter is the 



more intelligible. We shall limit our attention to the 

 right-hand half, as the algebraic expressions can be con- 

 tinuous only for the space between two points of applica- 

 tion of forces. 



Then the angular momentum round the point p pro- 

 duced by the force - is — X x, tending to throw the 



point of the bar upwards. 



The angular momentum in the opposite direction, 

 produced by the elasticity at p, is proportional to 



, or (if the flexure is not very large) 



,, C being the coefficient 



d'-y 



zC 



To determine 



'hen x = 



radius ot curvature at p 



to d J* ; and may be called C. £* 



</.r- d.\ 



to wdiich allusion is made above. 



Therefore C. -, ? must = - > 

 ax- 2 



Integrating, y — — x x* + constant 

 d x 4 c 



the constant, we remark that, when .1- = a the curve is 



parallel to the line a, or ' -' is o ; and therefore — X — 

 dx 4 C 4 



+ constant = o, or constant = - — X — : andthecom- 



4C 4 



plete value of -/ = — X( x- - — J. Integrating again, 

 dx 4C V 4/ fa 



y = — -_ X (' " ) + new constant. Wb 



4<- ^ 3 4 ' 



y must = o ; this gives new constant = -f-_: and the 



12 



complete value of y = — X ( A ' a ' A + - V This is 

 4C V3 4 12/ 



•W d 1 



to equal v when .r = o, or - - X — = b ; lrom which 

 4 C 12 



we obtain C = - 



48 . b 



(Second). Theory of the same bar, at least slightly 

 curved, in a vertical position ; its lower end supported on 

 1 he ground, &c, and its upper end loaded with a 

 weight //". 



It will be convenient here to take the centre of length 

 of the vertical line for origin of x. As no force or fixation 

 occurs between the two ends of the bar, the same theory 

 will apply throughout. 



Here the angular momentum of the weight W on the 

 point/, tending to bend the top to the right, is IV X y. 

 The angular momentum produced by the curvature at />, 



tending to throw the top to the left, is - C—£. Jltmay 



be convenient to remember that ' ~ is here a negative 

 dx- 



quantity!. To make these balance we have 



/,-,•= -c. ';■>:, or ''/-:: +i/:,'=o. 



ax- ax" L 



The solution of this equation isy = E . sin (xtj— ) + 



.F.cos (.rV", ) : where E and j^must be determined to 



suit the peculiarities of the case. Now, neglecting the 

 weight of the bar (which may usually be clone), the 

 curve will be symmetrical above and below; and there- 

 fore the value of y will be the same for x = + e and 



