ON THE INFLUENCE OF FORM ON STRENGTH. 427 



spring to receive accumulated work may be represented by the triangle fig. 16, 

 where a b represents the extension, and the several horizontal lines represent 

 the strains due to the different extensions, the strains increasing from at 

 " « " to 6 w ; and in order that the accumiilated work represented by the 

 parallelogram A B W may be transferred to the sirring, its extension must 

 be such that the area of the triangle a b lu shall equal that of the parallelo- 

 gram A BW. The maximum strain brought on the spring under these cir- 

 cumstances Avill clearly be no more than that represented by the length of 

 the horizontal line b iv. 



Now assume that the area of A B W has to be transferred to the four 

 springs D' of fig. 14, it is clear that only one-fourth of the area wiU have 

 to be borne by each spring, and the triangle representing the extension and 

 strain of each of the springs Avill only have one-fourth of the area of that of 

 a b IV. 



Let fig. 1 7 represent such a triangle, then, in order that its area may be 

 one -fourth, it follows that its sides must each be half of those of the tri- 

 angle a b w, that is, the length of a I, the extension, will be half of « b, and 

 the length of I m, the final strain, will be half that of b w ; but if this be true 

 of each of the springs D', the aggregate strain on the four springs must be 

 double that of the strain on D. 



But this double strain has, in the case of fig. 14, to be put on to the four 

 springs D' by means of the short single spring E, therefore this spring, which 

 is equal to D, will be put to a strain twice as much as that put uponD. 



It may be well to remark, in passing, that the fact of the ultimate strains 

 put on in arresting the accumulated work being double in the ease of fig. 14 

 to those of fig. 13, although the weight is the same in both cases, is by no 

 means inconsistent with the fact that when the springs are settled to rest 

 and are supporting W as a quiescent load, the sum of the strains of the four 

 -Springs D' must exactly equal that of the single spring D. 



It has not been thought necessary to take into account the small increase 

 in the faU of the weight W, due to the lowering of the collars B B' on the ex- 

 tension of the springs. 



There now comes to be considered the third proposition, that change of 

 form produces increased strain iinder impact, when the weight and inertia 

 of the object suffering the impact are taken into account. 



It is quite certain that in practice, where a falling weight is arrested by a 

 coUar B or B', the accumulated work of that weight would be partly taken up 

 by the elasticity of the bar D or D', and would be partly taken up by the 

 setting of the particles of the bar itself into motion, such motion being 

 .greatest at the bottom of the bar, and diminishing in the higher parts, until 

 ;at the top of the bar it would become nothing. 



Eeveiting to figs. 13, 14, 16, & 17, let it be assumed, for the sake of 

 simplicity, and as an illustration only, that all the weight of the apparatus 

 resides in the collars B B', and that the collar B' of the four-spring arrange- 

 ment, fig. 14, is four times as heavy as the collar B of the single-spring 

 arrangement, fig. 13. 



Further, reverting to fig. 16, the weight W would be brought to rest in the 

 .time during which it would traverse with a decreasing velocity the height a b, 

 and in fig. 17 the weight would be brought to rest in the height al, half of 

 that of a 6 ; and the time to bring the' weight W to rest in this latter case of 

 fig. 17 would be half that required to bring it to rest in fig. 16. 



Following this out, if the lines a b, a I be divided into the same number 

 of equal parts, say ten each, then the time to travel over part 1 in fig. 17 



2f-2 



