332 THE MATHEMATICAL PRINCIPLES [BOOK Jl the point I let there be drawn the right line KL parallel to the horizon and meeting the ice on both the sides thereof in K and L. Then the ve locity of the water running out at the hole EF will be the same that it would acquire by falling from I through the space IG. Therefore, by Galih'cJ's Theorems, IG will be to IH in the duplicate ratio of the velo city of the water that runs out at the hole to the velocity of the wrater in the circle AB, that is, in the duplicate ratio of the circle AB to the circle EF ; those circles being reciprocally as the velocities of the water which in the same time and in equal quantities passes severally through each of them, and completely fills them both. We are now considering the velo city with which the water tends to the plane of the horizon. But the mo tion parallel to the same, by which the parts of the falling water approach to each other, is not here taken notice of; since it is neither produced by gravity, nor at all changes the motion perpendicular to the horizon which the gravity produces. We suppose, indeed, that the parts of the water cohere a little, that by their cohesion they may in falling approach to each othei with motions parallel to the horizon in order to form one single cataract, and to prevent their being divided into several : but the motion parallel to the horizon arising from this cohesion does not come under our present consideration. CASE 1. Conceive now the w^hole cavity in the vessel, wrhich encompasses the falling water ABNFEM, to be full of ice, so that the water may pass through the ice as through a funnel. Then if the water pass very near to the ice only, without touching it; or, which is the same tiling, if by rea son of the perfect smoothness of the surface of the ice, the water, though touching it. glides over it writh the utmost freedom, and without the le-ast resistance; the water will run through the hole EF with the same velocity as before, and the whole weight of the column of water ABNFEM will be all taken up as before in forcing out the water, and the bottom of the vessel will sustain the weight of the ice encompassing that column. Let now the ice in the vessel dissolve into water ; yet will the efflux of the water remain, as to its velocity, the same as before. It will not be less, because the ice now dissolved will endeavour to descend ; it will not be greater, because the ice. now become water, cannot descend without hin dering the descent of other water equal to its own descent. The same force ought always to generate the same velocity in the effluent water. But the hole at the bottom of the vessel, by reason of the oblique mo tions of the particles of the effluent water, must be a little greater than before, For now the particles of the water do not all of them pass through the hole perpendicularly, but, flowing down on all parts from the sides of the vessel, and converging towards the hole, pass through it with oblique mo tions : r,r,d in tending downwards meet in a stream whose diameter is a little smaller below the hole than at the hole itself : its diameter being to the