320

MISCELLANEOUS

TlvACT 3S.

a blow, tbe ram falls from the height a + d, and therebysinks the pile the space b suppose; hence, for the next stroke,the fall will be a + d + b = a' in the theorem above, andd' + i-b’ =.■ d + b + ib', the next penetration or sinking being¥ ; the ref. a d : a d b : : {d ±b)b : ( d -|- b + -\¥)b ,a proportion which gives the quadratic equa. ¥ , ’+2¥{d+b)-=z

b)b, the root of which is ¥ = — (d + b) +

a+d+b / _ j |

— x (2 d

v[(d + by +

a + d + b . . .■ 7 , 7 -, a + d + b

( 2 d + b)b] = - a -~~ *

a + d

nearly, or indeed - ^■r-^-b nearly, because b is small in com-

~ 7 a + o *

parison with a + d.

Now, for an example in numbers, suppose a — 5 feet =60 inches, d =10 , b — 3, that is a — 60 the height of theram above the top of the pile before this enters the ground ;d — 10, after being fixed in the ground; and b = 3 thesinking by the next blow : then x 3 = 2'65:= ¥

o J d o 13

the 2d stroke. Next, substitutingd + b for d, and ¥ for b , the sametheorem gives 2"48 for the nextsinking, or the next value of ¥.

And so on continually, by whichmeans the series of the successivecorresponding values of the letterswill be as in the margin, the lastcolumn showing the several suc-cessive sinkings of the pile by therepeated strokes of the ram.

d + ihd + b

Specimen of the Seriesof the Successive va-lues of d, b, ¥.

d

b

¥

10

3

2-65

13

2 - 65

2-48

15-65

2-49

2-32

18-14

2-32

2-19

2046

&c.

2-19

2-08

Scholium. Thus then it appears, that the effect of anyoperation of pile-driving may be determined. It is manifestalso that the greater a is, or the higher the top of the machineis where the ram falls from, above the top of the pile at first,the greater will be every stroke of the ram, and consequentlythe fewer the strokes requisite to drive the pile to the requi-site depth. But then every stroke will take a longer time,as the ram will be both longer in falling and longer in rais-