hence vv = 2v, or v = of the wheel equal only to 3v, and v=V, or the velocity of the velocity of the water. PROBLEM VII. To determine the Form and Dimensions of Gunpowder Magazines. In the practice of engineering, with respect to the erection of powder magazines, the exterior shape is usually made like the roof of a house, having two sloping sides, forming two inclined planes, to throw off the rain, and meeting in an angle or ridge at the top; while the interior represents a vault, more or less extended, as the occasion may require; and the shape, or transverse section, in the form of some arch, both for strength and commodious room, for placing the powder barrels. It has been usual to make this interior curve a semicircle. But, against this shape, for such a purpose, I must enter my decided protest; as it is an arch the farthest of any from being in equilibrium in itself, and the weakest of any, by being unavoidably much thinner in one part than in others. Besides, it is constantly found, that after the centering of semicircular arches is struck, and removed, they settle at the crown, and rise up at the flanks, even with a straight horizontal form at top, and still much more so in powder magazines with a sloping roof; which effects are exactly what might be expected from a contemplation of the true theory of arches. Now this shrinking of the arches must be attended with other additional bad effects, by breaking the texture of the cement, after it has been in some de gree dried, and also by opening the joints of the voussoirs at one end. Instead of the circular arch therefore, we shall in this place give an investigation, founded on the true principles of equilibrium, of the only just form of the interior, which is properly adapted to the usual sloped roof. For this purpose, put a DK the thickness of the arch at the top, x = any absciss DP of the required arch ADCM, U KR the corresponding absciss of the given exterior line KI, and y = PC RI their equal ordinates. Then by the principles of arches, in my tracts on that subject, it is found that cr or to a + x ÿ x − x ÿ τ or Q X supposing a constant quantity, and where a is some certain quantity to be determined hereafter. But KR or u isty, if t be put to denote the the tangent of the given angle of elevation KIR, to radius 1, and then the equation is w= a + x = Qx ty= : but at D the value of w isa, and at D being parallel to KI; therefore the = 0, the curve correct fluent is the correct fluent of which gives ya x hyp. log. of w+ √(x2-22) Now, to determine the value of a, we are to consider that when the vertical line cr is in the position AL or MN, then w= ct becomes = AL or MN the given quantity e suppose, and y AQ or aмb suppose, in which position the c+(c2-a2) Last equation becomes bax hyp. log. (^2-22); and hence it is found that the value of the constant quantity vo, is which being substituted for it, in b h. 1.c+(-) the above general value of y, that value becomes a from which equation the value of the ordinate PC may always be found, to every given value of the vertical cı. 1 b But if, on the other hand, PC be given, to find CI, which will be the more convenient way, it may be found in the following manner: Put A = log. of a, and c = xlog. of G+ √(e2-a2) ; then the above equation gives cy + A= log. of we + √(x2-a); again, put = the number whose log. is cy + A; then n = w + √(w2 - a2); and hence w 2n a = CI. Now, for an example in numbers, in a real case of this nature, nature, let the foregoing figure represent a transverse vertical section of a magazine arch balanced in all its parts, in which the span or width AM is 20 feet, the pitch or height na is 10 feet, thickness at the crown DK = 7 feet, and the angle of the ridge LKN 112° 37′, or the half of it LKD = 56° 18′, the complement of which, or the elevation KIR, is 33° 41', the tangent of which is, which will therefore be the value of t in the foregoing investigation. The values of the other letters will be as follows, viz, DK=α=7; aq=b=10; DQ=h=10; AL=c=103: 3; A=log. of 7=8450980; log. of 2.562070408591; cy + A = ·0408591y + 8450980 log. of n. From the general equation then, viz, a2 + n2 = 2n a2 2n +, by assuming y successively equal to 1, 2, 3, 4, &c, thence finding the corresponding values of cy + ▲ or *0408591y + ·8450980, and to these, as common logs. taking out the corresponding natural numbers, which will be the values of n; then the above theorem will give the several values of or ci, as they are here arranged in the annexed table, from which the figure of the curve is to be constructed, by thus finding so many points in it. 8 9.0781 9 9.6628 10 10.3333 Otherwise. Instead of making n the number of the log. cy + A, if we put m = the natural number of the log. `w + √(w2 — n2) cy only; then m = and am—w=√(w2 — a2), or by squaring, &c, am2-2amw+w2 = w2 — a2, and hence xa; to which the numbers being applied, the very same conclusions result as in the foregoing calculation and table. w= m2 + 1 2m a PROBLEM VIII. To construct Powder Magazines with a Parabolical Arch. It has been shown, in my tract on the Principles of Arches of Bridges, that a parabolic arch is an arch of equilibration, when its extrados, or form of its exterior covering, is the very same parabola as the lower or inside curve. Hence then a parabolic arch, both for the inside and outer form, will be very Again, in prop. id. 1 1. re diversa na calculations of a cent Mr. Robins, or dermag he which balls re projected an powder. The dal i au meme. ball is discharged to a re wood, whose mal vedcir be easily observed md xmmer this small velocity, tins surned the pea e immediately derives y tus ante mov weight of the bayi ne am and the block, & i te u Hi Baur proportional, aici i de enam e evident that this simple note û eten of numerous sell me, a B... VEIC experiments as nate, van de m with all kinds and sms that the la the various sorts me manner TOM JE 1 experiment will say T excepting the extent of bar targe termine the resistance here. the block of word a differes sinave T completed in my dev printing. some results of the same pe ALATU applied to determine the suga force or elasticity of the ar pedung V der, and the venchy wih ata are circumstances which it were welon with any precision. A koum a other bulubky a sal be said, have mit Huset 2., tater til Geichstat Luca, That ingenious philosopher Da stupa kisprin showed that by die ingos parce o quitpower tity of elastie aur disengage wird, wuel collect in the space only super by the powder before it was footy was found to be dar 100 lines stronger that the weight or He then heated elasticity of the common atmospuene air. the same parcel of air IT the degree of red hot iron, auf found it in that temperature to be about 4 times as strong as before; whence be inferred, that the first strength of the sho Samed |