HYDRODYNAMICS.] Therefore G/x sin(0-) = < HYDROMECHANICS 457 = c tf ?>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 "y of rotation = l turn in n calibres. ., " 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 "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 * < 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", 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 (<? + 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 <f> 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 <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> 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-< a surface tension T at the common surface of the liquids.
"Tr .-Q
