TRACT 38.

PROBLEMS.

319

Now it was found above that y is as the penetration byany blow of the ram, by the same pile in the same soil, thatis, as the height fallen directly, and as the resistance or fric-tion in the earth inversely. Let a' denote any other and greaterheight, by an after stroke, and/' its friction ; also p the pe-netration by the former blow, and p' that by the latter, whichmust be the smaller: then, by the foregoing principle,

y: y : : p : p' •, hence a : a : : pf : pf, which is a generaltheorem.

But now, with respect to the quantity of friction from anyblow, though it be not known from experiment that the fric-tion is exactly proportional to the rubbing surface, there isgreat reason to believe that it must be at least very nearlyso : there is also equal reason to conclude that the effect orresistance from that rubbing surface must be nearly or ex-actly as the length of space it moves over, that is, by thepenetration of the pile by any blow. Now, if d denote thedepth of the pile in the ground before any new blow is struckby the ram, and b the depth or penetration produced by theblow, then the length of the rubbing surface will be d + -lb ;for, the length of the rubbing surface is only d at the begin-ning of the motion, and it is d + b at the end of it, the me-dium of the two', or d -)- \b, is therefore the due length ofthe surface, and the space or depth it moves over is b ; there-fore the whole resistance from the friction is (d + ib)b. Ifd' then denote any other depth of the pile in the earth, andb' the next penetration, then (d' -f lb')b' will bu its friction.Substituting now b for p, and b' for p , also d + ^-b for/, andd' + lb' for/', in the general theorem a : a! :\pf: pf', it be-comes a : a’ : : {d-\r\b)b : ( d'-\-±b')b ', for the general relationbetween the heights fallen and the resistance and penetra-tion.

This theorem will very conveniently give the series of ef-fects, or successive sinkings of the piles, by the blows of theram. Thus, after the pile has been properly fixed, or indeeddriven to any depth in the earth, denoted by d, then to give