452 HYDROMECHANICS [HYDRODYNAMICS. mass of a liquid - times the volume of the cylinder ; this is b- a-* called the effective inertia of the cylinder. In particular, if i = oc, the effective inertia is the mass of the cylinder, increased by a mass of liquid of equal volume with the cylinder ; and then i|/ = - V sin 6, <f> - V - i so that Ex, 2. The moving cylinder an elliptic cylinder, and the fixed cylinder a confocal elliptic cylinder. Using elliptic coordinates |, 77, such that c cosh 77, c sinh ri are the semi-axes of the confocal ellipse, c cos |, c sin of the confocal hyberbola passing through a point, 2c being the distance between the foci ; thena; = c cosh 77 cos|, y = c .sinh 77 sin f; and if rj = a is the equation of the moving ellipse, 77 = $ of the fixed ellipse, then satisfies the conditions that sinii (fi -a) (ii) i|/= - Vc sinh a sin = - Vy, when 7j=>-a, (iii) 4 / = ^> when 77 = 3- Therefore the conjugate function -, 7 . , cosh (3 - 77) d> = Vc sinh a - cos : sum ,.8 - a) 30 that L + id>*=Nc din 1 cos (t + it] - iff). sinh (3 - a) If <$> denote the velocity function of the liquid filling the elliptic cylinder 77 = a, then <f> = Va: = Vc cosh 77 cos | ; and round the ellipse 77 = a, <t> _ tanh a <f> tanh (3 - a) while is the same for each, and - vanishes when rj = 3; there - dn du fore the kinetic energy of the liquid between 77 = a and 77 = 3 is an *" a of the kinetic energy of the liquid inside TJ = a, which is tanh (3 - a) |rrpV 2 c 2 sinh a cosh a. Hence the mass of the cylinder 77 = 0; must be increased by au 1 a times the mass of an equal volume of liquid to give tanh (3 -a) the effective inertia for motion in the direction of the major axis, the space between the cylinder 77 = a and a fixed cylinder 77 = 3 being filled with liquid. Similarly for motion parallel to the minor axis, -, T , sinh (3 - T?) i = Vc cosh a ^ cos . sinh (3 -a) sinh (3 -a) (Quarterly Journal of Mathematics, vol. xvi.). Ex. 3. When the moving and fixed cylinders are any two circular cylinders, not co-axial, the limiting points are taken as the foci of reference ; and, supposing 2c the distance between them, and |, 77 the dipolar system of coordinates, we have sinh TJ sin | cosh 77 - cos | and then cosh TJ - cos = J_ i 0(r (x + c}~ + ?y 2 2 (x- cf + y 2 --tan" x-c x + c so that = constant is the equation of a circle passing through the two limiting points, and 77 = constant is the equation of an ortho gonal circle. If 77 = a be the moving cylinder, moving in the direction of the axis of x (the line of centres) with velocity V, and if 77 = 18 be the fixed cylinder, we must make sinh a + constant,
cosh a - c >s when 77 --a ; fy = Q when 7j = 3 ; and + = in the intervenin
tor various expressions for iff, consult the articles liy Mr W. M. Hicks in the Qtii -tcrly Journal of Mathematics, vol. xvi Now, expanding, sinh a cosh a - cos _ 1 = sinh 7^(77 - fl) a and therefore and ,. ,.. sinh n^a - @) Similarly for a velocity V of the cylinder 77^ a perpendicular to the line of centres, the cylinder 77 = 3 being fixed, sinh n(a- sinh n(a - 3) Next, suppose a rigid cylindrical surface to be rotating about the axis of z with angular velocity co ; we must have - = velocity of ds boundary normal to itself , dx dy ds ulj ds and therefore if/ =4 co(x 2 + ?/ 2 ) + constant, at all points of the moving boundary, and if/ = constant, at all points of a fixed cylindrical boundary. Ex. 4. Take the two elliptic cylinders of Ex. 2, and suppose the cylinder 77 = a to be rotating with angular velocity to, and the cylinder 77 = 3 to be fixed ; since ^ 2 + 2/ 2 = ~h c ~ (cosh 2TJ + COS 2|), if we put sinh 2(3 -a) then (i) when 77 = a , ty=wc- cos 2 = (a(x- + y~) + constant ; (ii) when TJ = 3 , ty = i (iii) Vo + i~^ = > a- WTJ J and therefore if/ satisfies the required conditions. T lipn <f> j&?c" sin 2, tnnrf)fQ olllll fJ CCy and from the value of the kinetic energy of the intermediate liquid the instantaneous value of the effective moment of inertia can be inferred. If the cylinder TJ = 3 be also rotating with angular velocity o>, the cylinders will remain confocal, and the values of ty and ^ will not change ; then ! wc2 sinh 2(7? - a) + sinh 2(3 - 77) cog ^ sinh 2(3 - a) (27j - a cosh (3 -a) cos 2 ; and coslM3-a) To find the kinetic energy of the liquid, since therefore and when therefore TJ = a, and TJ = 3, <p -5 = ^o> 2 c 4 tanh (3 - a) sin 2 2 ; T= ipa> 2 c 4 tanh (3 - a) /" sin 2 2{<Z "^ A = ^Trparc 4 tanh (3 - a) ; and, if Tc denote the effective radius of gyration of the liquid, T = ^7rpo; 2 c 2 & 2 (sinh 3 cosh j8-sinh a cosh a) ; tanh (3 - a) therefore sinh 3 cosh jS sinh a cosh a fC (aa 1 - Wi)(i&i -ab) where a,, b l are the semi-nxes of the ellipse Tj = 3, and a, b of the ellipse TJ = a. Ex. 5. Suppose a sector, bounded byr = aand 0=a, rotating
about the axis with angular velocity u> ; we must put
