In 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).

If 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.

I +W a W a), ' (k) 4 (I I) I+ w- R For a shot in air the ratio W'/**W i**s 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.

At 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.

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