426 REPOKT— 1869. 



of sucli a construction would be to pull down the hinged bar B, no longer in 

 a manner to preserve its horizontality, but to bring it down in a bent form, 

 and thereby to cause less of the load to be carried by the middle lower 

 spring C than its fair share of one-third, as carried by the middle spring of 

 fig. 9, and to throw of necessity this deficiency as an extra burthen upon the 

 two outer springs C C. 



The writer trusts he has succeeded in making his meaning clear, and has 

 proved his proposition that, even under a quiescent load, a sudden increase of 

 dimension in a suspended bar carrying such a load is a source of weakness. 

 If the writer has not succeeded in making his meaning clear, he is at a loss 

 how to illustrate it further, unless it be by some such proposition as this. 

 Assume 100 men opposed in line to 100 men also in line, then the conflict 

 would be equal for all; and then assume that one of the contending parties is 

 increased to 102 men by placing one man on each flank ; the two men will 

 clearly exert but little influence iipon those who are fighting in the centre of 

 the lines, but their presence wiU be most injuriously felt by the two flank men 

 of their opponents, as these two men will each have to contend with two 

 adversaries in lieu of one. In the same way the internal elastic columns 

 in the lower part of the bar fig. 8 have only similar fibres to deal with up 

 above ; but the external fibres of the lower part have not only to deal with 

 their own proper continuations above, but have also to deal with the fibres 

 above them in the parts projecting at the sides. 



The second proposition comes now to be considered, namely, the influence 

 of the abrupt change of form when the force of impact has to be resisted by 

 the elasticity of an assumed imponderable bar. 



Let D, fig. 11, be a plain bar of uniform section in its body part, but with 

 an enlargement at the top to enable it to be suspended, and with one at the 

 bottom to receive a collar, B, on which the weight, W, is supposed to fall 

 through the distance A B. 



The bar D', fig. 12, is assumed to have in its body part four times the sec- 

 tional area of D, but to terminate at its bottom end in a short piece E, 

 having a sectional area equal to D, or one quarter that of the upper part, D'. 

 This bar, D', is also provided with a weight, "\Y, falling through A B on to the 

 collar B'. 



Now let the spring arrangements of figs. 13 & 14 be substituted for the 

 bars D and D'. 



In fig. 13 there is a single spring D suspended from a support, and carrying 

 a collar B on which the weight W can strike on falling through the distance 

 AB. 



In fig. 14 there arc four suspended springs, D' D' D' D', each equal to D of 

 fig. 13. These springs are siipposed to be united at their lower ends to an 

 absolutely inflexible bar X, below which is a short single spring E, similar 

 in area and strength, so far as it goes, to either of the springs D, D'. 



This spring, E, supports the collar B', to be struck by the weight W on fall- 

 ing through the distance A B equal to the A B of fig. 13. 



Now if, as in any of these figs. 11 to 14, the weight W be suff'ered to fall 

 through the distance A B, the accumulated work residing in it when it 

 reaches B wiU equal the weight into the distance. 



Let this accumulated work be represented by the parallelogram (fig. 15) 

 ABAA^W. 



This accumulated work it is intended, in the case of fig. 13, to transfer to 

 the spring D, by the extension of that spring ; but as the resistance ofi'ered 

 by the spring will increase directly as the extension of it, the efficacy of the 



