Page:Encyclopædia Britannica, Ninth Edition, v. 12.djvu/473

457 HYDRODYNAMICS.] Therefore G/x sin(0-) = &amp;lt; HYDROMECHANICS 457 = c tf ?&amp;gt;i(sin - cos 6 tan ) = c?t sin 9 - c(j? sin cos -- ( c i ~ c tan ; and, dropping the factor sin 0, a quadratic equation in p, the condition for steady motion. The least admissible value of r in order that the roots should be real is given by In an oblate solid of revolution c x - c 3 is negative, and the roots of the quadratic in fj. are always real for all values of r. In a prolate solid c a -c 3 is positive, and a certain spin r is required to keep the motion stable. An interesting application is to determine the proper amount of rifling of a gun. The following table has been calculated, from the formula: given below, by Captain J. P. Cundill, R.A., and the re sults appear to agree very fairly with what is observed in practice. Table calculated for Stability of Rotation of Projectiles. _2 Minimum twist at muzzle of gun requisite to give stability &quot;y of rotation = l turn in n calibres. ., &quot; 3 Length of prc in oitlibres= Value of a-y. Cast-iron com mon shell. Cavity=j s ,ths vol. of sfiell. Density of iron = 7-207. I alliser shell. Cavity = Jth vol. of shell. Density = 8-000. Solid steel bullet. Density = 8-000. Solid lead and tin bullets of similar composi tion to M.-II. bullets. Density = 10-9. Value of n. Value of n. Value of n. Value of n. 2-0 49418 63-87 71-08 72-21 84-29 2-1 52032 59-84 66-59 67-66 78-98 2-2 54431 56-31 62-67 63 67 74-32 2 3 56643 53-19 59-19 60-14 70 20 2-4 58679 50-41 56-10 57 GO 66-53 2-5 60561 47-91 53-32 54-17 63-24 2-6 62315 45-65 50-81 51-62 60-26 27 63938 43-61 48-53 49-30 57 55 2-8 65454 41-74 46-45 47-19 55 -09 2-9 66368 40-02 44-54 45-25 52-72 3-0 68192 38-45 4279 43-47 50-74 8-1 69434 36-99 41-16 41-82 48-82 3-2 70598 35 64 39-66 40-30 47-04 33 71693 34-39 38-27 38-84 45-38 3-4 7-2724 33 &quot;22 36-97 37-56 43-84 3-5 -73697 32-13 3575 36 -33 42-40 3-6 74615 31-11 34-62 35-17 41-05 3-7 75483 3015 33-55 34-09 39-79 3-8 -76303 29-25 32-55 33 -07 38-61 3 9 77032 28-40 31-61 32-11 37-48 4-0 77820 27-60 30-72 31-21 36-43 Suppose the rifling at the muzzle makes one turn in n calibres, and la is the calibre and the angle of the rifling ; then tan = * r - = rt - = 2 /. r 3/ c _ c n u- / I Cj 1 * &amp;lt; If V = weight of shot, and W = weight of air displaced, then fi = W + V a, c 3 = W + V 7 , r, = W; 2 + W A^V, c (i = Vk&quot;, where /.-,, / are the radii of gyration of the shot about an equatorial diameter and the axis, and k[ of the air displaced, supposed rigidi- fied, about an equatorial axis; and then o, y, a will be certain quan tities depending only upon the external shape of the projectile, supposing the surrounding medium friction less and incom pressible. When, as in practice, the fraction ^- is so small that its square .may be neglected, . W. W (a w + W R = 4^(a-7)^ + higher powers of- which are neglected. The only body for which a, y, and o have as yet been determined is the ellipsoid; and in the case of a prolate spheroid of semi-axes a and c, = A~+C 7= 2A / (C - A)( c 2-a 2 ) 2 j (C - A)(c 2 - a 2 ) + (2A + C)(c 2 + a 2 ) j (&amp;lt;? + a?) where los C =

o (a 2 + A)(c . c+V(c 2 -a 2 (c 2 - a 2 )? e c - V(c 2 - a 2 ) c(c 2 - a 2 ) and therefore 2 A + C = -s- ere Wave Motion in Liquids. First consider plane waves propagated in the direction of the axis of x in liquid of depth h, the undisturbed surface being taken as the plane of xy and the axis of z drawn vertically upwards. The equation of continuity, supposing a velocity function &amp;lt;f&amp;gt; to exist, being tf+tf* n ~J~5 + -}- = , dx* dz* we must first seek a solution of this equation, involving a periodic term of the form sin (mx-nt), where z =, n = , being A. A the wave length and V the velocity of propagation of the waves. If we put =/(~) sin (mx - nt), then the solution of which, under the condition that -? = Q when dz 2= -h, is f(z) A cosh m(z + h), and therefore &amp;lt;p= A cosh m(z + h) sin (mx- nt). We must now endeavour to make the free surface a surface of equal pressure, and in order to do this we must suppose A small enough for its square to be neglected ; and therefore the square of the velocity is to be neglected too. The dynamical equation then becomes p d(f&amp;gt; p + + dt and at the surface where z = = II, a constant ; ,- = 0, and - may be put = - dt - dt - dz therefore g + = 0, when c = 0. dz dt 2 Amy sinh nth - n-A. cosh mh = n- = mg tanli mh, Therefore If the depth h be very great compared with the wave length A f then neglecting the square of , If the depth h be very small compared with the wave length A, then, neglecting the square of , A Next consider the more general case of wave motion propagated in the direction of the axis of ./ at the common surface z = of two liquids, the lower of density p and bounded below by the fixed plane z - h, and the upper of density p and bounded by the fixed plane z h , and suppose U and U the moan velocities of currents in the liquids making nng cs a .-ni l a with the axis of r ; suppose in addition there i-&amp;lt; a surface tension T at the common surface of the liquids. &quot;Tr .-Q