50. The theory preceding is of practical application in the investigation of the stability of the axial motion of a submarine boat, of the elongated gas bag of an airship, or of a spinning rifled projectile. In the steady motion under no force of such a body in a medium, the centre of gravity describes a helix, while the axis describes a cone round the direction of motion of the centre of gravity, and the couple causing precession is due to the displacement of the medium.
In the absence of a medium the inertia of the body to translation is the same in all directions, and is measured by the weight W, and under no force the C.G. proceeds in a straight line, and the axis of rotation through the C.G. preserves its original direction, if a principal axis of the body; otherwise the axis describes a cone, right circular if the body has uniaxial symmetry, and a Poinsot cone in the general case.
But the presence of the medium makes the effective inertia depend on the direction of motion with respect to the external shape of the body, and on W′ the weight of fluid medium displaced.
Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented byc1= W + W′α, c2= W + W′β,(1)where α, β are numerical factors depending on the external shape; and if the C.G. is moving with velocity V at an angle φ with the axis, so that the axial and broadside component of velocity is u = V cos φ, v = V sin φ, the total momentum F of the medium, represented by the vector OF at an angle θ with the axis, will have components, expressed in sec. ℔,F cos θ = c1u= (W + W′α)Vcos φ, F sin θ = c2v= (W + W′β)V.gggg(2)Suppose the body is kept from turning as it advances; after t seconds the C.G. will have moved from O to O′, where OO′ = Vt; and at O′ the momentum is the same in magnitude as before, but its vector is displaced from OF to O′F′.For the body alone the resultant of the components of momentumWVcos φ and WVsin φ is WVsec. ℔,ggg(3)acting along OO′, and so is unaltered.But the change of the resultant momentum F of the medium as well as of the body from the vector OF to O′F′ requires an impulse couple, tending to increase the angle FOO′, of magnitude, in sec. foot-poundsF·OO′·sin FOO′ = FVt sin (θ − φ),(4)equivalent to an incessant coupleN = FV sin (θ − φ)= (F sin θ cos φ − F cos θ sin φ) V= (c2− c1) (V2/ g) sin φ cos φ= W′ (β − α) uv / g.(5)This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c2− c1is positive to set the body broadside to the advance.An oblate flattened body, like a disk or plate, has c2− c1negative, so that the medium steers the body axially; this may be verified by a plate dropped in water, and a leaf or disk or rocket-stick or piece of paper falling in air. A card will show the influence of the couple N if projected with a spin in its plane, when it will be found to change its aspect in the air.An elongated body like a ship has c2− c1positive, and the couple N tends to disturb the axial movement and makes it unstable, so that a steamer requires to be steered by constant attention at the helm.Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination θ. With no reserve of buoyancy W = W′, and the couple N, tending to increase θ, has the effect of diminishing the metacentric height by h ft. vertical, whereWh tan θ = N = (c2− c1)c1u2tan θ,c2g(6)h =c2− c1c1u2= (β − α)1 + αu2.Wc2g1 + βg(7)
Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented by
c1= W + W′α, c2= W + W′β,
(1)
where α, β are numerical factors depending on the external shape; and if the C.G. is moving with velocity V at an angle φ with the axis, so that the axial and broadside component of velocity is u = V cos φ, v = V sin φ, the total momentum F of the medium, represented by the vector OF at an angle θ with the axis, will have components, expressed in sec. ℔,
(2)
Suppose the body is kept from turning as it advances; after t seconds the C.G. will have moved from O to O′, where OO′ = Vt; and at O′ the momentum is the same in magnitude as before, but its vector is displaced from OF to O′F′.
For the body alone the resultant of the components of momentum
(3)
acting along OO′, and so is unaltered.
But the change of the resultant momentum F of the medium as well as of the body from the vector OF to O′F′ requires an impulse couple, tending to increase the angle FOO′, of magnitude, in sec. foot-pounds
F·OO′·sin FOO′ = FVt sin (θ − φ),
(4)
equivalent to an incessant couple
N = FV sin (θ − φ)= (F sin θ cos φ − F cos θ sin φ) V= (c2− c1) (V2/ g) sin φ cos φ= W′ (β − α) uv / g.
N = FV sin (θ − φ)
= (F sin θ cos φ − F cos θ sin φ) V
= (c2− c1) (V2/ g) sin φ cos φ
= W′ (β − α) uv / g.
(5)
This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c2− c1is positive to set the body broadside to the advance.
An oblate flattened body, like a disk or plate, has c2− c1negative, so that the medium steers the body axially; this may be verified by a plate dropped in water, and a leaf or disk or rocket-stick or piece of paper falling in air. A card will show the influence of the couple N if projected with a spin in its plane, when it will be found to change its aspect in the air.
An elongated body like a ship has c2− c1positive, and the couple N tends to disturb the axial movement and makes it unstable, so that a steamer requires to be steered by constant attention at the helm.
Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination θ. With no reserve of buoyancy W = W′, and the couple N, tending to increase θ, has the effect of diminishing the metacentric height by h ft. vertical, where
(6)
(7)
51. An elongated shot is made to preserve its axial flight through the air by giving it the spin sufficient for stability, without which it would turn broadside to its advance; a top in the same way is made to stand upright on the point in the position of equilibrium, unstable statically but dynamically stable if the spin is sufficient; and the investigation proceeds in the same way for the two problems (seeGyroscope).
The effective angular inertia of the body in the medium is now required; denote it by C1about the axis of the figure, and by C2about a diameter of the mean section. A rotation about the axis of a figure of revolution does not set the medium in motion, so that C1is the moment of inertia of the body about the axis, denoted by Wk12. But if Wk22is the moment of inertia of the body about a mean diameter, and ω the angular velocity about it generated by animpulsecouple M, and M′ is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k′,Wk22ω = M − M′, W′k′2ω = M′,(1)(Wk22+ W′k′2) ω = M,(2)C2= Wk22+ W′k′2= (W + W′ε) k22,(3)in which we have put k′2= εk2, where ε is a numerical factor depending on the shape.If the shot is spinning about its axis with angular velocity p, and is preceding steadily at a rate μ about a line parallel to the resultant momentum F at an angle θ, the velocity of the vector of angular momentum, as in the case of a top, isC1pμ sinθ − C2μ2sin θ cos θ;(4)and equating this to the impressed couple (multiplied by g), that is, togN = (c1− c2)c1u2tan θ,c2(5)and dividing out sin θ, which equated to zero would imply perfect centring, we obtainC2μ2cos θ − C1pμ + (c2− c1)c1u2sec θ = 0.c2(6)The least admissible value of p is that which makes the roots equal of this quadratic in μ, and thenμ = ½C1p sec θ,C2(7)the roots would be imaginary for a value of p smaller than given byC12p2− 4 (c2− c1)c1C2u2= 0,c2(8)p2= 4 (c2− c1)c1C2.u2c2C12(9)Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving δ the Angle of Rifling, and n the Pitch of Rifling in Calibres.Cast-iron Common Shellƒ =2⁄3, S.G. 7.2.Palliser Shellƒ = ½, S.G. 8.Solid Steel Bulletƒ = 0, S.G. 8.Solid Lead Bulletƒ = 0, S.G. 10.9.xβ − αδnδnδnδn1.00.00000° 0′Infinity0° 0′Infinity0° 0′Infinity0° 0′Infinity2.00.49422 4963.872 3271.082 2972.212 0884.292.50.60563 4647.913 2353.323 1954.172 5163.243.00.68194 4138.454 1342.794 0943.473 3850.743.50.73705 3532.135 0235.754 5836.334 1542.404.00.77826 3027.605 5130.725 4531.214 5636.434.50.81007 2424.206 4026.936 3227.365 3731.945.00.83518 1621.567 2823.987 2124.366 1828.446.00.872110 0517.679 0419.678 5619.987 4023.3310.00.939516 5710.3115 1911.4715 0511.6513 0013.60Infinity1.000090 000.0090 000.0090 000.0090 000.00If 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 δ of the rifling is given bytan δ = πd / nd = ½ dp / u,(10)which is the ratio of the linear velocity of rotation ½dp to u, the velocity of advance,tan2δ =π2=d2p2= (c2− c1)c1C2d2n24u2c2C12=W′(β − α)1 +W′αW·(1 +W′ε)(k1)2Wd.W1 +W′βW(k1)4W(11)For a shot in air the ratio W′/W is so small that the square may be neglected, and formula (11) can be replaced for practical purpose in artillery bytan2δ =π2=W′(β − α)(k2)2/(k1)4,n2Wdd(12)if then we can calculate β, α, or β − α for the external shape of the shot, this equation will give the value of δ and n required for stability of flight in the air.The ellipsoid is the only shape for which α and β have so far been determined analytically, as shown already in § 44, so we must restrict our calculation to an egg-shaped bullet, bounded by a prolate ellipsoid of revolution, in which, with b = c,A0=∫∞0ab2dλ=∫∞0ab2dλ,(a2+ λ) √ [ 4 (a2+ λ) (b2+ λ)2]2 (a2+ λ)3/2(b2+ λ)(13)A0+ 2B0= 1,(14)a =A0, β =B0=1 − A0=1.1 − A01 − B01 + A01 + 2α(15)The length of the shot being denoted by l and the calibre by d, and the length in calibres by xl / d = 2a / 2b = x,(16)A0=xch−1x −1,(x2− 1)3/2x2− 1(17)2B0=− xch−1x +x2,(x2− 1)3/2x2+ 1(18)x2A0+ 2B0=x sh−1√ (x2− 1)=xlog [ x + √ (x2− 1) ].√ (x2− 1)√ (x2− 1)(19)If σ 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 being1⁄6π d3, and1⁄6π d3x of a shot x calibres longW =1⁄6πd3x (i − ƒ3) σ,(20)Wk12=1⁄6πd3xd2(1 − ƒ5) σ,10(21)Wk22=1⁄6πd3xl2+ d2(1 − ƒ5) σ.20(22)If ρ denotes the density of the air or mediumW′ =1⁄6πd3xρ,(23)W′=1ρ,W1 − ƒ3σ(24)k12=11 − ƒ5,k22=x2+ 1,d2101 − ƒ3k122(25)tan2δ =ρ(β − α)x2+ 1,σ1⁄5(1 − ƒ5)(26)in which σ/ρ may be replaced by 800 times the S.G. of the metal, taking water as 800 times denser than air on the average, in round numbers, and formula (10) may be written n tan δ = π, or nδ = 180, when δ is a small angle, and given in degrees.From this formula (26) the table following has been calculated by A. G. Hadcock, and the results are in agreement with practical experience.52. In the steady motion the centre of the shot describes a helix, with axial velocityu cos θ = v sin θ =(l +c1tan2θ)u cos θ ≈ u sec θ,c2(1)and transverse velocityu sin θ − v cos θ =(l −c1)u sin θ ≈ (β − α) u sin θ;c2(2)and the time of completing a turn of the spiral is 2π/μ.When μ has the critical value in (7),2π=4πC2cos θ =2π(x2+ 1) cos θ,μpC1p(3)which makes the circumference of the cylinder on which the helix is wrapped2π(u sin θ − v cos θ =2πu(β − α) (x2+ 1) sin2θ cos θμp= nd (β − α) (x2+ 1) sin θ cos θ,(4)and the length of one turn of the helix2π(u cos θ + v sin θ) = nd (x2+ 1);μ(5)thus for x = 3, the length is 10 times the pitch of the rifling.53.The Motion of a Perforated Solid in Liquid.—In the preceding investigation, the liquid stops dead when the body is brought to rest; and when the body is in motion the surrounding liquid moves in a uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on its surface is the opposite of that required to change the motion of the liquid; this has been expressed by the dynamical equations given above. But if the body is perforated, the liquid can circulate through a hole, in reentrant stream lines linked with the body, even while the body is at rest; and no reaction from the surface can influence this circulation, which may be supposed started in the ideal manner described in § 29, by the application of impulsive pressure across an ideal membrane closing the hole, by means of ideal mechanism connected with the body. The body is held fixed, and the reaction of the mechanism and the resultant of the impulsive pressure on the surface are a measure of the impulse, linear ξ, η, ζ, and angular λ, μ, ν, required to start the circulation.This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences an additional reaction from the circulation which is the opposite of the force required to change the position in space of the circulation impulse; and these extra forces must be taken into account in the dynamical equations.An article may be consulted in thePhil. Mag., April 1893, by G. H. Bryan, in which the analytical equations of motion are deduced of a perforated solid in liquid, from considerations purely hydrodynamical.The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by ξ the resultant axial linear momentum of the circulation.As the ring is moved from O to O′ in time t, with velocity Q, and angular velocity R, the components of liquid momentum change fromαM′U + ξ and βM′V along Ox and OytoαM′U′+ ξ and βM′V′ along O′x′ and O′y′,(1)the axis of the ring changing from Ox to O′x′; andU = Q cos θ, V = Q sin θ,U′ = Q cos (θ − Rt), V′ = Q sin (θ − Rt),(2)so that the increase of the components of momentum, X1, Y1, and N1, linear and angular, areX1= (αM′U′ + ξ) cos Rt − αM′U − ξ − βM′V′ sin Rt=(α − β)M′Q sin (θ − Rt) sin Rt − ξ ver Rt(3)Y1= (αM′U′ + ξ) sin Rt + βM′V′ cos Rt − βM′V= (α − β) M′Q cos (θ − Rt) sin Rt + ξ sin RT,(4)N1= [ −(αM′U′ + ξ) sin (θ − Rt) + βM′V′ cos (θ − Rt) ] OO′= [ −(α − β) M′Q cos (θ − Rt) sin (θ − Rt) − ξ sin (θ − Rt) ] Qt.(5)The components of force, X, Y, and N, acting on the liquid at O, and reacting on the body, are thenX = lt. X1/t = (α − β) M′QR sin θ = (α − β) M′VR,(6)Y = lt. Y1/t = (α − β) M′QR cos θ + ξR = (α − β) M′UR + ξR,(7)Z = lt. Z1/t = −(α − β) M′Q2sin θ cos θ − ξQ sin θ = [ −(α − β) M′U + ξ ] V.(8)Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid−αM′(dU− VR), −βM′(dV+ UR), εC′dR,dtdtdt(9)so that its equations of motion areM(dU− VR)= −αM′(dU− VR)− (α − β) M′VR,dtdt(10)M(dV+ UR)= −βM′(dV+ UR)− (α − β) M′UR − ξR,dtdt(11)CdR= −εC′dR+ (α − β) M′UV + ξV;dtdt(12)and putting as beforeM + αM′ = c1, M + βM′ = c2, C + εC′ = C3,(13)c1dUc2VR = 0,dt(14)c2dV+ (c1U + ξ) R = 0,dt(15)c3dR− (c1U + ξ − c2U) V = 0;dt(16)showing the modification of the equations of plane motion, due to the component ξ of the circulation.The integral of (14) and (15) may be writtenc1U + ξ = F cos θ, c2V = − F sin θ,(17)dx= U cos θ − V sin θ =F cos2θ+F sin2θ−ξcos θ,dtc1c2c1(18)dμ= U sin θ + V cos θ =(F−F)sin θ cos θ −ξsin θ,dtc1c2c1(19)C3d2θ=(F2−F2)sin θ cos θ −Fξsin θ = Fdμ,dt2c1c2c1dt(20)C3dθ= Fy =√ [−F2cos2θ−F2sin2θ+ 2Fξcos θ + H];dtc1c2c1(21)so that cos θ and y is an elliptic function of the time.When ξ is absent, dx/dt is always positive, and the centre of the body cannot describe loops; but with ξ, the influence may be great enough to make dx/dt change sign, and so loops occur, as shown in A. B. Basset’sHydrodynamics, i. 192, resembling the trochoidal curves, which can be looped, investigated in § 29 for the motion of a cylinder under gravity, when surrounded by a vortex.The branch of hydrodynamics which discusses wave motion in a liquid or gas is given now in the articlesSoundandWave; while the influence of viscosity is considered underHydraulics.References.—For the history and references to the original memoirs seeReport to the British Association, by G. G. Stokes (1846), and W. M. Hicks (1882). See also theFortschritte der Mathematik, and A. E. H. Love, “Hydrodynamik” in theEncyklöpadie der mathematischen Wissenschaften(1901).
The effective angular inertia of the body in the medium is now required; denote it by C1about the axis of the figure, and by C2about a diameter of the mean section. A rotation about the axis of a figure of revolution does not set the medium in motion, so that C1is the moment of inertia of the body about the axis, denoted by Wk12. But if Wk22is the moment of inertia of the body about a mean diameter, and ω the angular velocity about it generated by animpulsecouple M, and M′ is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k′,
Wk22ω = M − M′, W′k′2ω = M′,
(1)
(Wk22+ W′k′2) ω = M,
(2)
C2= Wk22+ W′k′2= (W + W′ε) k22,
(3)
in which we have put k′2= εk2, where ε is a numerical factor depending on the shape.
If the shot is spinning about its axis with angular velocity p, and is preceding steadily at a rate μ about a line parallel to the resultant momentum F at an angle θ, the velocity of the vector of angular momentum, as in the case of a top, is
C1pμ sinθ − C2μ2sin θ cos θ;
(4)
and equating this to the impressed couple (multiplied by g), that is, to
(5)
and dividing out sin θ, which equated to zero would imply perfect centring, we obtain
(6)
The least admissible value of p is that which makes the roots equal of this quadratic in μ, and then
(7)
the roots would be imaginary for a value of p smaller than given by
(8)
(9)
Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving δ the Angle of Rifling, and n the Pitch of Rifling in Calibres.
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 δ of the rifling is given by
tan δ = πd / nd = ½ dp / u,
(10)
which is the ratio of the linear velocity of rotation ½dp to u, the velocity of advance,
(11)
For a shot in air the ratio W′/W is so small that the square may be neglected, and formula (11) can be replaced for practical purpose in artillery by
(12)
if then we can calculate β, α, or β − α for the external shape of the shot, this equation will give the value of δ and n required for stability of flight in the air.
The ellipsoid is the only shape for which α and β have so far been determined analytically, as shown already in § 44, so we must restrict our calculation to an egg-shaped bullet, bounded by a prolate ellipsoid of revolution, in which, with b = c,
(13)
A0+ 2B0= 1,
(14)
(15)
The length of the shot being denoted by l and the calibre by d, and the length in calibres by x
l / d = 2a / 2b = x,
(16)
(17)
(18)
(19)
If σ 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 being1⁄6π d3, and1⁄6π d3x of a shot x calibres long
W =1⁄6πd3x (i − ƒ3) σ,
(20)
(21)
(22)
If ρ denotes the density of the air or medium
W′ =1⁄6πd3xρ,
(23)
(24)
(25)
(26)
in which σ/ρ may be replaced by 800 times the S.G. of the metal, taking water as 800 times denser than air on the average, in round numbers, and formula (10) may be written n tan δ = π, or nδ = 180, when δ is a small angle, and given in degrees.
From this formula (26) the table following has been calculated by A. G. Hadcock, and the results are in agreement with practical experience.
52. In the steady motion the centre of the shot describes a helix, with axial velocity
(1)
and transverse velocity
(2)
and the time of completing a turn of the spiral is 2π/μ.
When μ has the critical value in (7),
(3)
which makes the circumference of the cylinder on which the helix is wrapped
= nd (β − α) (x2+ 1) sin θ cos θ,
(4)
and the length of one turn of the helix
(5)
thus for x = 3, the length is 10 times the pitch of the rifling.
53.The Motion of a Perforated Solid in Liquid.—In the preceding investigation, the liquid stops dead when the body is brought to rest; and when the body is in motion the surrounding liquid moves in a uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on its surface is the opposite of that required to change the motion of the liquid; this has been expressed by the dynamical equations given above. But if the body is perforated, the liquid can circulate through a hole, in reentrant stream lines linked with the body, even while the body is at rest; and no reaction from the surface can influence this circulation, which may be supposed started in the ideal manner described in § 29, by the application of impulsive pressure across an ideal membrane closing the hole, by means of ideal mechanism connected with the body. The body is held fixed, and the reaction of the mechanism and the resultant of the impulsive pressure on the surface are a measure of the impulse, linear ξ, η, ζ, and angular λ, μ, ν, required to start the circulation.
This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences an additional reaction from the circulation which is the opposite of the force required to change the position in space of the circulation impulse; and these extra forces must be taken into account in the dynamical equations.
An article may be consulted in thePhil. Mag., April 1893, by G. H. Bryan, in which the analytical equations of motion are deduced of a perforated solid in liquid, from considerations purely hydrodynamical.
The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by ξ the resultant axial linear momentum of the circulation.
As the ring is moved from O to O′ in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from
αM′U + ξ and βM′V along Ox and Oy
to
αM′U′+ ξ and βM′V′ along O′x′ and O′y′,
(1)
the axis of the ring changing from Ox to O′x′; and
U = Q cos θ, V = Q sin θ,U′ = Q cos (θ − Rt), V′ = Q sin (θ − Rt),
U = Q cos θ, V = Q sin θ,
U′ = Q cos (θ − Rt), V′ = Q sin (θ − Rt),
(2)
so that the increase of the components of momentum, X1, Y1, and N1, linear and angular, are
X1= (αM′U′ + ξ) cos Rt − αM′U − ξ − βM′V′ sin Rt=(α − β)M′Q sin (θ − Rt) sin Rt − ξ ver Rt
X1= (αM′U′ + ξ) cos Rt − αM′U − ξ − βM′V′ sin Rt
=(α − β)M′Q sin (θ − Rt) sin Rt − ξ ver Rt
(3)
Y1= (αM′U′ + ξ) sin Rt + βM′V′ cos Rt − βM′V= (α − β) M′Q cos (θ − Rt) sin Rt + ξ sin RT,
Y1= (αM′U′ + ξ) sin Rt + βM′V′ cos Rt − βM′V
= (α − β) M′Q cos (θ − Rt) sin Rt + ξ sin RT,
(4)
N1= [ −(αM′U′ + ξ) sin (θ − Rt) + βM′V′ cos (θ − Rt) ] OO′= [ −(α − β) M′Q cos (θ − Rt) sin (θ − Rt) − ξ sin (θ − Rt) ] Qt.
N1= [ −(αM′U′ + ξ) sin (θ − Rt) + βM′V′ cos (θ − Rt) ] OO′
= [ −(α − β) M′Q cos (θ − Rt) sin (θ − Rt) − ξ sin (θ − Rt) ] Qt.
(5)
The components of force, X, Y, and N, acting on the liquid at O, and reacting on the body, are then
X = lt. X1/t = (α − β) M′QR sin θ = (α − β) M′VR,
(6)
Y = lt. Y1/t = (α − β) M′QR cos θ + ξR = (α − β) M′UR + ξR,
(7)
Z = lt. Z1/t = −(α − β) M′Q2sin θ cos θ − ξQ sin θ = [ −(α − β) M′U + ξ ] V.
(8)
Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid
(9)
so that its equations of motion are
(10)
(11)
(12)
and putting as before
M + αM′ = c1, M + βM′ = c2, C + εC′ = C3,
(13)
(14)
(15)
(16)
showing the modification of the equations of plane motion, due to the component ξ of the circulation.
The integral of (14) and (15) may be written
c1U + ξ = F cos θ, c2V = − F sin θ,
(17)
(18)
(19)
(20)
(21)
so that cos θ and y is an elliptic function of the time.
When ξ is absent, dx/dt is always positive, and the centre of the body cannot describe loops; but with ξ, the influence may be great enough to make dx/dt change sign, and so loops occur, as shown in A. B. Basset’sHydrodynamics, i. 192, resembling the trochoidal curves, which can be looped, investigated in § 29 for the motion of a cylinder under gravity, when surrounded by a vortex.
The branch of hydrodynamics which discusses wave motion in a liquid or gas is given now in the articlesSoundandWave; while the influence of viscosity is considered underHydraulics.
References.—For the history and references to the original memoirs seeReport to the British Association, by G. G. Stokes (1846), and W. M. Hicks (1882). See also theFortschritte der Mathematik, and A. E. H. Love, “Hydrodynamik” in theEncyklöpadie der mathematischen Wissenschaften(1901).
(A. G. G.)
HYDROMEDUSAE,a group of marine animals, recognized as belonging to the Hydrozoa (q.v.) by the following characters. (1) The polyp (hydropolyp) is of simple structure, typically much longer than broad, without ectodermal oesophagus or mesenteries, such as are seen in the anthopolyp (see articleAnthozoa); the mouth is usually raised above the peristome on a short conical elevation or hypostome; the ectoderm is without cilia. (2) With very few exceptions, the polyp is not the only type of individual that occurs, but alternates in the life-cycle of a given species, with a distinct type, the medusa (q.v.), while in other cases the polyp-stage may be absent altogether, so that only medusa-individuals occur in the life-cycle.
The Hydromedusae represent, therefore, a sub-class of the Hydrozoa. The only other sub-class is the Scyphomedusae (q.v.). The Hydromedusae contrast with the Scyphomedusae in the following points. (1) The polyp, when present, is without the strongly developed longitudinal retractor muscles, forming ridges (taeniolae) projecting into the digestive cavity, seen in the scyphistoma or scyphopolyp. (2) The medusa, when present, has a velum and is hence said to becraspedote; the nervous system forms two continuous rings running above and below the velum; the margin of the umbrella is not lobed (except in Narcomedusae) but entire; there are characteristic differences in the sense-organs (see below, andScyphomedusae); and gastral filaments (phacellae), subgenital pits, &c., are absent. (3) The gonads, whether formed in the polyp or the medusa, are developed in the ectoderm.
The Hydromedusae form a widespread, dominant and highly differentiated group of animals, typically marine, and found in all seas and in all zones of marine life. Fresh-water forms, however, are also known, very few as regards species or genera, but often extremely abundant as individuals. In the British fresh-water fauna only two genera,HydraandCordylophora, are found; in America occurs an additional genus,Microhydra. The paucity of fresh-water forms contrasts sharply, with the great abundance of marine genera common in all seas and on every shore. The species ofHydra, however, are extremely common and familiar inhabitants of ponds and ditches.
In fresh-water Hydromedusae the life-cycle is usually secondarily simplified, but in marine forms the life-cycle may be extremely complicated, and a given species often passes in the course of its history through widely different forms adapted to different habitats and modes of life. Apart from larval or embryonic forms there are found typically two types of person, as already stated, the polyp and the medusa, each of which may vary independently of the other, since their environment and life-conditions are usually quite different. Hence both polyp and medusa present characters for classification, and a given species, genus or other taxonomic category may be defined by polyp-characters or medusa-characters or by both combined. If our knowledge of the life-histories of these organisms were perfect, their polymorphism would present no difficulties to classification; but unfortunately this is far from being the case. In the majority of cases we do not know the polyp corresponding to a given medusa, or the medusa that arises from a given polyp.1Even when a medusa is seen to be budded, from a polyp under observation in an aquarium, the difficulty is not always solved, since the freshly-liberated, immature medusa may differ greatly from the full-grown, sexually-mature medusa after several months of life on the high seas (see figs. 11, B, C, and 59,a,b,c). To establish the exact relationship it is necessary not only to breed but to rear the medusa, which cannot always be done inconfinement. The alternative is to fish all stages of the medusa in its growth in the open sea, a slow and laborious method in which the chance of error is very great, unless the series of stages is very complete.
At present, therefore, classifications of the Hydromedusae have a more or less tentative character, and are liable to revision with increased knowledge of the life-histories of these organisms. Many groups bear at present two names, the one representing the group as defined by polyp-characters, the other as defined by medusa-characters. It is not even possible in all cases to be certain that the polyp-group corresponds exactly to the medusa-group, especially in minor systematic categories, such as families.
The following is the main outline of the classification that is Adopted in the present article. Groups founded on polyp-characters are printed in ordinary type, those founded on medusa-characters in italics. For definitions of the groups see below.
Sub-class Hydromedusae (Hydrozoa Craspedota).Order I. Eleutheroblastea.” II. Hydroidea (Leptolinae).Sub-order 1. Gymnoblastea (Anthomedusae).” 2. Calyptoblastea (Leptomedusae).Order III. Hydrocorallinae.” IV. Graptolitoidea.” V. Trachylinae.Sub-order 1.Trachomedusae.” 2.Narcomedusae.Order VI. Siphonophora.Sub-order 1. Chondrophorida.” 2. Calycophorida.” 3. Physophorida.” 4. Cystophorida.
Sub-class Hydromedusae (Hydrozoa Craspedota).
Order I. Eleutheroblastea.
” II. Hydroidea (Leptolinae).
Sub-order 1. Gymnoblastea (Anthomedusae).
” 2. Calyptoblastea (Leptomedusae).
Order III. Hydrocorallinae.
” IV. Graptolitoidea.
” V. Trachylinae.
Sub-order 1.Trachomedusae.
” 2.Narcomedusae.
Order VI. Siphonophora.
Sub-order 1. Chondrophorida.
” 2. Calycophorida.
” 3. Physophorida.
” 4. Cystophorida.
Organization and Morphology of the Hydromedusae.
a, Hydranth;
b, Hydrocaulus;
c, Hydrorhiza;
t, Tentacle;
ps, Perisarc, forming in the region of the hydranth a cup or hydrotheca(h,t),—which, however, is only found in polyps of the order Calyptoblastea.
As already stated, there occur in the Hydromedusae two distinct types of person, the polyp and the medusa; and either of them is capable of non-sexual reproduction by budding, a process which may lead to the formation of colonies, composed of more or fewer individuals combined and connected together. The morphology of the group thus falls naturally into four sections—(1) the hydropolyp, (2) the polyp-colony, (3) the hydromedusa, (4) the medusa-colonies. Since, however, medusa-colonies occur only in one group, the Siphonophora, and divergent views are held with regard to the morphological interpretation of the members of a siphonophore, only the first three of the above subdivisions of hydromedusa morphology will be dealt with here in a general way, and the morphology of the Siphonophora will be considered under the heading of the group itself.