SOD PHILOSOPHICAL TRANSACTIONS. [aNNO i77g. 



posed of the inverse ratios of tlie superficies and of the masses, or d : d z=: 



J 1 

 s X M '■ s X ni := : . 



s X m s X M 



If now all the circumstances be unequal, viz. the f;illing weights, the heights, 

 the weights of the piles, and their superficies in tlie earth; then the depths, 

 sunk at each stroke, will be in the ratio composed of the direct ratios of the 

 falling weights and altitudes, and of the inverse ratios of the superficies and 

 masses. Let us now suppose a 3d pile, its mass = m, the superficies sunk s, 

 the falling weight w, the altitufle fallen o, and the depth the pile sinks by a 

 given percussion ; then, by what is done above, it appears that 



d:S=-^:-^, and:D = a Xw.AXvf; therefore d:D = ^-^^ : ^ ^ ^ 

 mx^MXS _ m X s M X a 



These theorems will serve for practice, and for comparing the effects of different 

 pile-drivers. 



Problem. — To delermine the depth sunk by the pile at each stroke of the 

 ram. — Since in this case both the ram and the weight of the pile are constant 

 and equal, then vr ■= w and m = m. Hence the fundamental proportion will 

 be (/: D = - : -. And the superficies of the piles in the earth are rectangles of 

 the same base but of ditFerent altitude d and d (where d denotes the total depth). 

 Therefore «: s = t< : d', more simply and conveniently thus expressed c/ : d = 

 -^ : -,. After a few strokes, so as that the pile may stand firmly in the ground, 

 a new stroke is then given, which may be called the first, then the weight falls 

 from the altitude a, and the pile sinks by the depth d. A 2d stroke is then 

 given, by which the pile sinks the depth x; the ram falls through the altitude 

 A = a + f/, and the whole depth of the pile will be d = rf -f ^'- By substitu- 

 tion then we have d:x-=--.: j-—- . From which original analogy is deduced the 



following equation, x^ -\- dx = d^ -\ — -, and hence .r = + V'{-d' -f- — ) — 

 xd, the value of the unknown quantity. 



To apply this theorem to a given example: let the altitude fallen by the ram 

 be a = 3 feet = 36 inches; the depth sunk by the pile at the first stroke 

 <i = 4 inches; hence a == 2-|- inches nearly. Supposing this 2d stroke to sink 

 the pile 3 inches in round numbers. Then in the 3d stroke the ram will fall 

 through 36 -1- 4 -f 3 = 43 inches; the depth of the pile will be 4 -f 3 = 7 

 inches: then the depth sunk by the 3d stroke will come out .?■ = 2 inches nearly. 

 For the 4th stroke the altitude fallen will be 36 -|- 4 + 3 -|- 2 = 45 inches, and 

 the depth of the pile 4 -f 3 + 2 = Q inches: therefore the depth sunk by the 

 4th stroke will be:r = l-i- inch nearly. 



Belidor also solves the same problem. He supposes tlie pile to sink at first 1 5 

 inches: then by his calculus it sinks at the 2d stroke 17. at the 3d ig, at the 



