372

MISCELLANEOUS

TRACT 38.

of, and produced by the force of gravity, as well as that ofany other falling body. Hence the height due to the velo-city, that is, the height to be freely fallen by any body toacquire the observed velocity of the natural stream, in theriver a little way above bridge, becomes known. From thesame velocity also will be found that of the increased currentin the narrowed way of the arches, by taking it in the reci-procal proportion of the breadth of the river above, to thecontracted way in the arches ; viz, by saying, as the latter isto the former, so is the first velocity, or slower motion, tothe quicker. Next, from this last velocity, will be foundthe height due to it as before, that is, the height to be freelyfallen through by gravity, to produce it. Then the differ-ence of these two heights, thus freely fallen by gravity, toproduce the two velocities, is the required quantity of thewaterfall in the arches; allowing however, in the calculation,for the contraction, in the narrowed passage, at the rate asobserved by Sir I. Newton, in prop. 36 of the 2d book of thePrincipia, or by other authors, being nearly in the ratio of25 to 21, or sufficiently near as 24 to 20, that is, as 6 to 5.

Such then are the elements and principles, on which thesolution of the problem is easily made out as follows.

Let b = the breadth of the channel in feet;

v — mean velocity of the water in feet per second.;c = breadth of the waterway between the obstacles.

Now 6 : 5 : : c : —c, the waterway contracted as above.

And — c : b :: v : — w, the velocity in the contracted way.Also 32 2 : v 2 : : 16 : -^v 2 , height fallen to gain the velocity®.And 32* : (-^ v) 2 :: 16 : (~Y x -&V 1 , ditto for the vel. ^ v.Then (~) 2 X — — is the measure of the fall required.Or [(I") 1- 1] X jjj is a rule for computing the fall.

Or rather — X v l very nearly, for the fall.