If W I is the weight of the train in pounds, the rate of working against the gradient expressed in horse-power units is H.P.=W,V/550 G.

00In order to obtain the seminvari ants we would write down the (w; 0, n) terms each associated with a literal coefficient; if we now operate with 52 we obtain a linear function of (w - I; 8, n) products, for the vanishing of which the literal coefficients must satisfy (w-I; 0, n) linear equations; hence (w; 8, n)-(w-I; 0, n) of these coefficients may be assumed arbitrarily, and the number of linearly independent solutions of 52=o, of the given degree and weight, is precisely (w; 8, n) - (w - I; 0, n).

00(9) c 2 Ci If the shot is moving as if fired from a gun of calibre d inches, in which the rifling makes one turn in a pitch of n calibres or nd inches, so that the angle S of the rifling is given by tan S = ird/nd = 2 d p/u, (10) '°If a denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal shape, a fraction f of the linear scale; then the volume of a round shot being sird 3, and sird 3 x of a shot x calibres long W =*ird 3 x(I -f 3)v, (20) 2 Wki 2= 61rd 3 xo(I-f 5)Q, (21) Wk22=67rd3x 2 2+0 2(I - f5)Q.

00If p denotes the density of the air or medium W' = sird 3 xp, (23) W' I p __ W I -1 3 k12 I k22 x2 ±i a 2= 101-1 3 '111 2= 2 tan g S = Q (l - a) x 2+ I (26) in which a/p may be replaced by 800 times the S.G.

00I +W a W a), ' (k) 4 (I I) I+ w- R For a shot in air the ratio W'/W is so small that the square may be neglected, and formula (II) can be replaced for practical purpose in artillery by tan26= n2 = W i (0 - a) (k ð)7()4, (12) if then we can calculate /3, a, or (3-a for the external shape of the shot, this equation will give the value of 6 and n required for stability of flight in the air.

00At the end of the 6th century the exarchate included Istria; the maritime part of Venetia as distinct from the interior which was in the hands of the Lombard kings at Pavia; the exarchate proper, or territory around Ravenna on the eastern side of the Apennines, to which was added Calabria, which at that period meant the heel and not the toe of the boot; the Pentapolis, or coast from Rimini to Ancona with the interior as far as the mountains; the duchy of Rome, or belt of territory connecting the Pentapolis with the western coast, the coast of Naples, w i th Bruttium the toe of the boot, the modern Calabria, and Liguria, or the Riviera of Genoa.

00pressure, W i the weight in water, atmospheric conditions remaining very nearly the same; p is the density of the water in which the body is weighed, a is (I +at°) in which a is the coefficient of cubical expansion of the body, and S is the density of the air at t°, p mm.

00Then W e = (W1+Wf)(k+k 2 +k 3 ...) will be the weight of main girders to carry W i -}-W f and their own weight (Buck, Proc. Inst.

00The greatest shear at C may occur with W I at C. If W 1 passes FIG.

0046) be the section at which the greatest shear is required, and let the loads advance from the left till W I is at C. If R is the resultant of the loads then on the girder, the reaction at B and shear at C is Rn/l.

00But the shear may be greater when W2 is at C. In that case the shear at C becomes R(n+a)/l+d - WI, if a>p, and R(n+a)/l+d - W I a/p, if a WI in the first case, and if Ra/l >W l al p in the second case.

00W-i. ? ?

00Now let the solid be placed in the lower pan, care being taken that no bubbles of air remain attached to it, and let w i be the weight now required in the scale pan..

00His work (` w i lacxa) in twenty-four books, written in Greek, is rather a number of monographs than a connected history.

00Fay l'tmos W i ideat ?OOU?A D?>:"

00(9) c 2 Ci If the shot is moving as if fired from a gun of calibre d inches, in which the rifling makes one turn in a pitch of n calibres or nd inches, so that the angle S of the rifling is given by tan S = ird/nd = 2 d p/u, (10) 'Ã‚°If a denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal shape, a fraction f of the linear scale; then the volume of a round shot being sird 3, and sird 3 x of a shot x calibres long W =*ird 3 x(I -f 3)v, (20) 2 Wki 2= 61rd 3 xo(I-f 5)Q, (21) Wk22=67rd3x 2 2+0 2(I - f5)Q.

00I +W a W a), ' (k) 4 (I I) I+ w- R For a shot in air the ratio W'/W is so small that the square may be neglected, and formula (II) can be replaced for practical purpose in artillery by tan26= n2 = W i (0 - a) (k Ã°)7()4, (12) if then we can calculate /3, a, or (3-a for the external shape of the shot, this equation will give the value of 6 and n required for stability of flight in the air.

00pressure, W i the weight in water, atmospheric conditions remaining very nearly the same; p is the density of the water in which the body is weighed, a is (I +atÃ‚°) in which a is the coefficient of cubical expansion of the body, and S is the density of the air at tÃ‚°, p mm.

00The bending moment at C is M = W I x(l - m)/l+W 2 m{I - (x+a)/l }.

00If the loads are moved a distance Ax to the right, the bending moment becomes M+OM =W I (x+Ax)(l - m)/l+W 2 m{1 - (x+Ox +a) /l} Om = WIDx(l - m)/1 - W20xm/l, and this is positive or the bending moment increases, if W I (l - m) >W 2 m, or if WI/m>W2/(l - m).

00

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