(25)an elliptic integral, of the third kind, with pole at z = E; and thenῶ − ψ = KCH = tan−1KH/CH= tan−1A sin θ dθ/dt= tan−1√ (2Z),G′ − G cos θ(G′ − Gz) / An(26)which determines ψ.Otherwise, from the geometry of fig. 4,C′K sin θ = OC − OC′ cos θ,(27)A sin2θ dψ/dt = G − G′ cos θ,(28)ψ =∫G − G′zdt= ½∫G − G′dt+ ½∫G + G′dt,1 − z2A1 − zA1 + zA(29)the sum of two elliptic integrals of the third kind, with pole at z = ±1; and the relation in (25) (26) shows the addition of these two integrals into a single integral, with pole at z = E.The motion of a sphere, rolling and spinning in the interior of a spherical bowl, or on the top of a sphere, is found to be of the same character as the motion of the axis of a spinning top about a fixed point.The curve described by H can be identified as a Poinsot herpolhode, that is, the curve traced out by rolling a quadric surface with centre fixed at O on the horizontal plane through C; and Darboux has shown also that a deformable hyperboloid made of the generating lines, with O and H at opposite ends of a diameter and one generator fixed in OC, can be moved so as to describe the curve H; the tangent plane of the hyperboloid at H being normal to the curve of H; and then the other generator through O will coincide in the movement with OC′, the axis of the top; thus the Poinsot herpolhode curve H is also the trace made by rolling a line of curvature on an ellipsoid confocal to the hyperboloid of one sheet, on the plane through C.Kirchhoff’sKinetic Analogueasserts also that the curve of H is the projection of a tortuous elastica, and that the spherical curve of C′ is a hodograph of the elastica described with constant velocity.Writing the equation of the focal ellipse of the Darboux hyperboloid through H, enlarged to double scale so that O is the centre,x2/α2+ y2/β2+ z2/O = 1,(30)with α2+ λ, β2+ λ, λ denoting the squares of the semiaxes of a confocal ellipsoid, and λ changed into μ and ν for a confocal hyperboloid of one sheet and of two sheets.λ > 0 > μ > −β2> ν > −α2,(31)then in the deformation of the hyperboloid, λ and ν remain constant at H; and utilizing the theorems of solid geometry on confocal quadrics, the magnitudes may be chosen so thatα2+ λ + β2+ μ + ν = OH2= ½k2(F − z) = ρ2+ OC2.(32)α2+ μ = ½k2(z1− z) = ρ2− ρ12,(33)β2+ μ = ½k2(z2− z) = ρ2− ρ22,(34)μ = ½k2(z3− z) = ρ2− ρ32,(35)ρ12< 0 < ρ22< ρ2< ρ32,(36)F = z1+ z2+ z3,(37)λ − 2μ + ν = k2z, λ − ν = k2,(38)λ − μ=1 + z,μ − ν=1 − zλ − ν2λ − ν2(39)with z = cos θ, θ denoting the angle between the generating lines through H; and with OC = δ, OC′ = δ′, the length k has been chosen so that in the preceding equationsδ/k = G/2An, δ′/k = G′/2An;(40)and δ, δ′, k may replace G, G′, 2An; then2Z=1(dθ)2=4KH2,1 − z2n2dtk2(41)while from (33-39)2Z=4 (α2+ μ) (β2+ μ) μ,1 − z2k2(μ − λ) (μ − ν)(42)which verifies that KH is the perpendicular from O on the tangent plane of the hyperboloid at H, and so proves Darboux’s theorem.Planes through O perpendicular to the generating lines cut off a constant length HQ = δ, HQ′ = δ′, so the line of curvature described by H in the deformation of the hyperboloid, the intersection of the fixed confocal ellipsoid λ and hyperboloid of two sheets ν, rolls on a horizontal plane through C and at the same time on a plane through C′ perpendicular to OC′.Produce the generating line HQ to meet the principal planes of the confocal system in V, T, P; these will also be fixed points on the generator; and putting(HV, HT, HP,)/HQ = D/(A, B, C,),(43)thenAx2+ By2+ Cz2= Dδ2(44)is a quadric surface with the squares of the semiaxes given by HV·HQ, HT·HQ, HP·HQ, and with HQ the normal line at H, and so touching the horizontal plane through C; and the direction cosines of the normal beingx/HV, y/HT, z/HP,(45)A2x2+ B2y2+ C2z2= D2δ2,(46)the line of curvature, called the polhode curve by Poinsot, being the intersection of the quadric surface (44) with the ellipsoid (46).There is a second surface associated with (44), which rolls on the plane through C′, corresponding to the other generating line HQ′ through H, so that the same line of curvature rolls on two planes at a constant distance from O, δ and δ′; and the motion of the top is made up of the combination. This completes the statement of Jacobi’s theorem (Werke, ii. 480) that the motion of a top can be resolved into two movements of a body under no force.Conversely, starting with Poinsot’s polhode and herpolhode given in (44) (46), the normal plane is drawn at H, cutting the principal axes of the rolling quadric in X, Y, Z; and thenα2+ μ = x·OX, β2+ μ = y·OY, μ = z·OZ,(47)this determines the deformable hyperboloid of which one generator through H is a normal to the plane through C; and the other generator is inclined at an angle θ, the inclination of the axis of the top, while the normal plane or the parallel plane through O revolves with angular velocity dψ/dt.The curvature is useful in drawing a curve of H; the diameter of curvature D is given byD =dp2=½k2sin3θ,½D=¼k2.dpδ − δ′pKM·KN(48)The curvature is zero and H passes through a point of inflexion when C′ comes into the horizontal plane through C; ψ will then be stationary and the curve described by C′ will be looped.In a state of steady motion, z oscillates between two limits z2and z3which are close together; so putting z2= z3the coefficient of z in Z is2Z1z3+ z23= −1 +GG′= −1 +(OM cos θ + ON) (OM + ON cos θ),A2n2OM·ON(49)2z1z3=OM2+ ON2cos θ, z1=OM2+ ON2,OM·ON2OM·ON(50)2 (z1− z3) =OM2− 2OM·ON cos θ + ON2=MN2.OM·ONOM·ON(51)With z2= z3, κ = 0, K = ½π; and the number of beats per second of the axis ism=n√z1− z3=MNn,ππ2√ (OM·ON)2π(52)beating time with a pendulum of lengthL =l=4OM·ONl.½ (z1− z3)MN2(53)The wheel making R/2π revolutions per second,beats/second=MNn=C·MN,revolutions/second√ (OM·ON)RAOC′(54)from (8) (9) § 3; and the apsidal angle isμ½π=Aμ·n·½π =ON·2√ (OM·ON)·½π =ONπ,mAnm√ (OM·ON)MNMN(55)and the height of the equivalent conical pendulum λ is given byλ=g=n2=OM=KC=OL,llμ2μ2ONKC′OC′(56)if OR drawn at right angles to OK cuts KC′ in R, and RL is drawn horizontal to cut the vertical CO in L; thus if OC2represents l to scale, then OL will represent λ.9. The gyroscope motion in fig. 4 comes to a stop when the rim of the wheel touches the ground; and to realize the motion when the axis is inclined at a greater angle with the upward vertical, the stalk is pivoted in fig. 8 in a lug screwed to the axle of a bicycle hub, fastened vertically in a bracket bolted to a beam. The wheel can now be spun by hand, and projected in any manner so as to produce a desired gyroscopic motion, undulating, looped, or with cusps if the stalk of the wheel is dropped from rest.As the principal part of the motion takes place now in the neighbourhood of the lowest position, it is convenient to measure the angle θ from the downward vertical, and to change the sign of z and G.Equation (18) § 8 must be changed tomt = nt√z3− z1=∫z3z√ (z3− z1) dz,2√ (4Z)(1)Z = (z − F) (1 − z2) − (G2− 2GG′z + G′2) / 2A2n2= (z − D) (1 − z2) − (G − G′z)2/ 2A2n2= (z − E) (1 − z2) − (G′ − Gz)2/ 2A2n2= (z3− z) (z − z2) (z − z1),(2)1 > z3> z > z2> −1, D, E > z1,(3)z1+ z2+ z3= F = D − G′2/ 2A2n2= E − G2/ 2A2n2,(4)and expressed by the inverse elliptic functionmt = sn−1√z3− z= cn−1√z − z2= dn−1√z − z1,z3− z2z3− z2z3− z1(5)z = z2sn2mt + z3cn2mt, κ2= (z3− z2) / (z3− z1).(6)Equation (25) and (29) § 8 is changed toῶ = ½∫G′ − Gzdt= ½∫G′ − GEdt−Gt,z − EAz − EA2A(7)ψ =∫G′z − Gdt= ½∫G′ + Gdt− ½∫G′ − Gdt,1 − z2A1 − zA1 + zA(8)while ψ and ῶ change places in (26).The Jacobian elliptic parameter of the third elliptic integral in (7) can be given by ν, wherev =∫z3E√ (z3− z1)dz =∫z3z2+∫z2E= K + (1 − f) Ki′,√ (4Z)(9)where f is a real fraction,(1 − f) K′ =∫z2E√ (z3− z1)dz,√ (−4Z)(10)fK′ =∫Ez1√ (z3− z1)dz,√ (−4Z)= sn−1√E − z1= cn−1√z2− E= dn−1√z3− E,z2− z1z2− z1z3− z1(11)with respect to the comodulus κ′.Then, with z = E, and2ZE= −{ (G′ − GE) / An}2,(12)if II denotes the apsidal angle of ῶ, and T the time of a single beat of the axle, up or down,II +GT=∫z3z2√ (−2ZE)dz,2Az − E√ (2Z)= ½πf + Kznf K′,(13)in accordance with the theory of the complete elliptic integral of the third kind.Interpreted geometrically on the deformablehyperboloid, flattened in the plane of the focal ellipse, if OQ is the perpendicular from the centre on the tangent HP, AOQ = amfK′, and the eccentric angle of P, measured from the minor axis, is am(1 − f) K′, the eccentricity of the focal ellipse being the comodulus κ′.A point L is taken in QP such thatQL/OA = znfK′,(14)QV, QT, QP = OA (zs, zc, zd) fK′;(15)and withmT = K, m/n = √ (z3− z1) /2 = OA/k,(16)GT=G·kK =QHK,2A2AnOAOA(17)II = ½πf +QL + QHK = ½πf +HLK.OAOA(18)By choosing for f a simple rational fraction, such as ½,1⁄3, ¼,1⁄5, ... an algebraical case of motion can be constructed (Annals of Mathematics, 1904).Thus with G′ − GE = 0, we have E = z1or z2, never z3; f = 0 or 1; and P is at A or B on the focal ellipse; and thenῶ = −pt, p = G/2A,(19)ψ + pt = tan−1n√ (2Z),2p (z − E)(20)sin θ exp (ψ + pt) i = i√ [(−z2− z3) (z − z1)] + √ [(z3− z) (z − z2)],z1=1 + z2z3,√−z2− z3=G=p=G′, orz2+ z322Ann2Anz1(21)sin θ exp (ψ + pt)i = i√ [(−z1− z3) (z − z2)] + √ [(z3− z) (z − z1)],z2=1 + z1z3,√−z1− z3=G=p=G′.z1+ z322Ann2Anz1(22)Thus z2= 0 in (22) makes G′ = 0; so that if the stalk is held out horizontally and projected with angular velocity 2p about the vertical axis OC without giving any spin to the wheel, the resulting motion of the stalk is like that of a spherical pendulum, and given bysin θ exp (ψ + pt)i = i√(2p2cos θ)+√(sin2θ − 2p2cos θ),n2n2= i sin α √ (sec α cos θ) + √ [(sec α + cos θ) (cos α − cos θ)],(23)if the axis falls in the lowest position to an angle α with the downward vertical.With z3= 0 in (21) and z2= −cos β, and changing to the upward vertical measurement, the motion is given bysin θ eψi= eint√ ½ cos β [√ (1 − cos β cos θ) + i√ (cos β cos θ − cos2θ)],(24)and the axis rises from the horizontal position to a series of cusps; and the mean precessional motion is the same as in steady motion with the same rotation and the axis horizontal.The special case of f = ½ may be stated here; it is found thatpexp (ῶ − pt) i =√(1 + x) (κ − x)+ i√(1 − x) (κ + x),a22(25)ρ2= a2(κ − x2),(26)½λ2sin θ exp (ψ − pt) i = (L − 1 + κ − x)√(1 − x) (κ + x2+ i (L − 1 + κ + x)√(1 + x) (κ − x),2(27)L = ½ (1 − κ) + λp/n,(28)so that p = 0 and the motion is made algebraical by taking L = ½ (1 − κ).The stereoscopic diagram of fig. 12 drawn by T. I. Dewar shows these curves for κ =15⁄17,3⁄5, and1⁄3(cusps).10. So far the motion of the axis OC’ of the top has alone been considered; for the specification of any point of the body, Euler’s third angle φ must be introduced, representing the angular displacement of the wheel with respect to the stalk. This is given bydφ+ cos θdψ= R,dtdt(1)d(φ + ψ)=(1 −C)R +G′ + G,dtAA (1 + cos θ)d(φ − ψ)=(1 −C)R +G′ − G.dtAA (1 − cos θ)(2)It will simplify the formulas by cancelling a secular term if we make C = A, and the top is then called aspherical top; OH becomes the axis of instantaneous angular velocity, as well as of resultant angular momentum.When this secular term is restored in the general case, the axis OI of angular velocity is obtained by producing Q′H to I, makingHI=A − C,HI=A − C,Q′HCQ′IA(3)and then the four vector components OC′, C′K, KH, HI give a resultant vector OI, representing the angular velocity ω, such thatOI/Q′I = ω/R.(4)Fig. 12.The point I is then fixed on the generating line Q′H of the deformable hyperboloid, and the other generator through I will cut the fixed generator OC of the opposite system in a fixed point O′, such that IO′ is of constant length, and may be joined up by a link, which constrains I to move on a sphere.In the spherical top then,½ (φ + ψ)=∫G′ + Gdt, ½ (φ − ψ)=∫G′ − Gdt1 + z2A1 − z2A(5)depending on the two elliptic integrals of the third kind, with pole at z = ±1; and measuring θ from the downward vertical, their elliptic parameters are:—v1=∫∞1√ (z3− z1) dz= f1K′i,√ (4Z)(6)v2=∫−1−∞√ (z3− z1) dzK + (1 − f2) K′i,√ (4Z)(7)f1K′ =∫∞1√ (z3− z1) dz√ ( −4Z)= sn−1√z3− z1= cn−1√1 − z3= dn−1√1 − z2,1 − z11 − z11 − z1(8)(1 − f2) K′ =∫−1z1√ (z3− z1) dz√ ( −4Z)= sn−1√−1 − z1= cn−1√1 + z2= dn−1√1 + z3.z2− z1z2− z1z3− z1(9)Then if v′ = K + (1 − f′)K′i is the parameter corresponding to z = D, we findf = f2− f1, f′ = f2+ f1,(10)v = v1+ v2, v′ = v1− v2.(11)The most symmetrical treatment of the motion of any point fixed in the top will be found in Klein and Sommerfeld, Theorie des Kreisels, to which the reader is referred for details; four new functions, α, β, γ, δ, are introduced, defined in terms of Euler’s angles, θ, ψ, φ, byα = cos ½θ exp ½ (φ + ψ) i,(12)β = i sin ½θ exp ½ (−φ + ψ) i,(13)γ = i sin ½θ exp ½ (φ − ψ) i,(14)δ = cos ½θ exp ½ (−φ − ψ) i.(15)Next Klein takes two functions or co-ordinates λ and Λ, defined byλ =x + yi=r + z,r − zx − yi(16)and Λ the same function of X, Y, Z, so that λ, Λ play the part of stereographic representations of the same point (x, y, z) or (X, Y, Z) on a sphere of radius r, with respect to poles in which the sphere is intersected by Oz and OZ.These new functions are shown to be connected by the bilinear relationλ =αΛ + β, αδ − βγ = 1,γΛ + δ(17)in accordance with the annexed scheme of transformation of co-ordinates—ΞΗΖξα2β22αβηγ2δ22γδζαγβδαδ + βγwhereξ = x + yi, η = −x + yi, ζ = −z,Ξ = X + Yi, Η = −X + Yi, Ζ = −Z;(18)and thus the motion in space of any point fixed in the body defined by Λ is determined completely by means of α, β, γ, δ; and in the case of the symmetrical top these functions are elliptic transcendants, to which Klein has given the name ofmultiplicative elliptic functions; andαδ = cos2½θ, βγ = −sin2½θ,αδ − βγ = 1, αδ + βγ = cos θ,√ ( −4αβγδ) = sin θ;(19)while, for the motion of a point on the axis, putting Λ = 0, or ∞,λ = β/δ = i tan ½θeψi, or λ = α/γ = −i cot ½θeψi,(20)andαβ = ½i sin θeψi, αγ = ½i sin θeψi,(21)giving orthogonal projections on the planes GKH, CHK; andαdβ−dαβ = nρeῶi,dtdtk(22)the vectorial equation in the plane GKH of the herpolhode of H for a spherical top.When f1and f2in (9) are rational fractions, these multiplicative elliptic functions can be replaced by algebraical functions, qualified by factors which are exponential functions of the time t; a series of quasi-algebraical cases of motion can thus be constructed, which become purely algebraical when the exponential factors are cancelled by a suitable arrangement of the constants.Thus, for example, with f = 0, f′ = 1, f1= ½, f2= ½, as in (24) § 9, where P and P′ are at A and B on the focal ellipse, we have for the spherical top(1 + cos θ) exp (φ + ψ − qt) i= √ (sec β − cos θ) √ (cos β − cos θ) + i (√ sec β + √ cos β) √ cos θ,
(25)
an elliptic integral, of the third kind, with pole at z = E; and then
ῶ − ψ = KCH = tan−1KH/CH
(26)
which determines ψ.
Otherwise, from the geometry of fig. 4,
C′K sin θ = OC − OC′ cos θ,
(27)
A sin2θ dψ/dt = G − G′ cos θ,
(28)
(29)
the sum of two elliptic integrals of the third kind, with pole at z = ±1; and the relation in (25) (26) shows the addition of these two integrals into a single integral, with pole at z = E.
The motion of a sphere, rolling and spinning in the interior of a spherical bowl, or on the top of a sphere, is found to be of the same character as the motion of the axis of a spinning top about a fixed point.
The curve described by H can be identified as a Poinsot herpolhode, that is, the curve traced out by rolling a quadric surface with centre fixed at O on the horizontal plane through C; and Darboux has shown also that a deformable hyperboloid made of the generating lines, with O and H at opposite ends of a diameter and one generator fixed in OC, can be moved so as to describe the curve H; the tangent plane of the hyperboloid at H being normal to the curve of H; and then the other generator through O will coincide in the movement with OC′, the axis of the top; thus the Poinsot herpolhode curve H is also the trace made by rolling a line of curvature on an ellipsoid confocal to the hyperboloid of one sheet, on the plane through C.
Kirchhoff’sKinetic Analogueasserts also that the curve of H is the projection of a tortuous elastica, and that the spherical curve of C′ is a hodograph of the elastica described with constant velocity.
Writing the equation of the focal ellipse of the Darboux hyperboloid through H, enlarged to double scale so that O is the centre,
x2/α2+ y2/β2+ z2/O = 1,
(30)
with α2+ λ, β2+ λ, λ denoting the squares of the semiaxes of a confocal ellipsoid, and λ changed into μ and ν for a confocal hyperboloid of one sheet and of two sheets.
λ > 0 > μ > −β2> ν > −α2,
(31)
then in the deformation of the hyperboloid, λ and ν remain constant at H; and utilizing the theorems of solid geometry on confocal quadrics, the magnitudes may be chosen so that
α2+ λ + β2+ μ + ν = OH2= ½k2(F − z) = ρ2+ OC2.
(32)
α2+ μ = ½k2(z1− z) = ρ2− ρ12,
(33)
β2+ μ = ½k2(z2− z) = ρ2− ρ22,
(34)
μ = ½k2(z3− z) = ρ2− ρ32,
(35)
ρ12< 0 < ρ22< ρ2< ρ32,
(36)
F = z1+ z2+ z3,
(37)
λ − 2μ + ν = k2z, λ − ν = k2,
(38)
(39)
with z = cos θ, θ denoting the angle between the generating lines through H; and with OC = δ, OC′ = δ′, the length k has been chosen so that in the preceding equations
δ/k = G/2An, δ′/k = G′/2An;
(40)
and δ, δ′, k may replace G, G′, 2An; then
(41)
while from (33-39)
(42)
which verifies that KH is the perpendicular from O on the tangent plane of the hyperboloid at H, and so proves Darboux’s theorem.
Planes through O perpendicular to the generating lines cut off a constant length HQ = δ, HQ′ = δ′, so the line of curvature described by H in the deformation of the hyperboloid, the intersection of the fixed confocal ellipsoid λ and hyperboloid of two sheets ν, rolls on a horizontal plane through C and at the same time on a plane through C′ perpendicular to OC′.
Produce the generating line HQ to meet the principal planes of the confocal system in V, T, P; these will also be fixed points on the generator; and putting
(HV, HT, HP,)/HQ = D/(A, B, C,),
(43)
then
Ax2+ By2+ Cz2= Dδ2
(44)
is a quadric surface with the squares of the semiaxes given by HV·HQ, HT·HQ, HP·HQ, and with HQ the normal line at H, and so touching the horizontal plane through C; and the direction cosines of the normal being
x/HV, y/HT, z/HP,
(45)
A2x2+ B2y2+ C2z2= D2δ2,
(46)
the line of curvature, called the polhode curve by Poinsot, being the intersection of the quadric surface (44) with the ellipsoid (46).
There is a second surface associated with (44), which rolls on the plane through C′, corresponding to the other generating line HQ′ through H, so that the same line of curvature rolls on two planes at a constant distance from O, δ and δ′; and the motion of the top is made up of the combination. This completes the statement of Jacobi’s theorem (Werke, ii. 480) that the motion of a top can be resolved into two movements of a body under no force.
Conversely, starting with Poinsot’s polhode and herpolhode given in (44) (46), the normal plane is drawn at H, cutting the principal axes of the rolling quadric in X, Y, Z; and then
α2+ μ = x·OX, β2+ μ = y·OY, μ = z·OZ,
(47)
this determines the deformable hyperboloid of which one generator through H is a normal to the plane through C; and the other generator is inclined at an angle θ, the inclination of the axis of the top, while the normal plane or the parallel plane through O revolves with angular velocity dψ/dt.
The curvature is useful in drawing a curve of H; the diameter of curvature D is given by
(48)
The curvature is zero and H passes through a point of inflexion when C′ comes into the horizontal plane through C; ψ will then be stationary and the curve described by C′ will be looped.
In a state of steady motion, z oscillates between two limits z2and z3which are close together; so putting z2= z3the coefficient of z in Z is
(49)
(50)
(51)
With z2= z3, κ = 0, K = ½π; and the number of beats per second of the axis is
(52)
beating time with a pendulum of length
(53)
The wheel making R/2π revolutions per second,
(54)
from (8) (9) § 3; and the apsidal angle is
(55)
and the height of the equivalent conical pendulum λ is given by
(56)
if OR drawn at right angles to OK cuts KC′ in R, and RL is drawn horizontal to cut the vertical CO in L; thus if OC2represents l to scale, then OL will represent λ.
9. The gyroscope motion in fig. 4 comes to a stop when the rim of the wheel touches the ground; and to realize the motion when the axis is inclined at a greater angle with the upward vertical, the stalk is pivoted in fig. 8 in a lug screwed to the axle of a bicycle hub, fastened vertically in a bracket bolted to a beam. The wheel can now be spun by hand, and projected in any manner so as to produce a desired gyroscopic motion, undulating, looped, or with cusps if the stalk of the wheel is dropped from rest.
As the principal part of the motion takes place now in the neighbourhood of the lowest position, it is convenient to measure the angle θ from the downward vertical, and to change the sign of z and G.
Equation (18) § 8 must be changed to
(1)
Z = (z − F) (1 − z2) − (G2− 2GG′z + G′2) / 2A2n2= (z − D) (1 − z2) − (G − G′z)2/ 2A2n2= (z − E) (1 − z2) − (G′ − Gz)2/ 2A2n2= (z3− z) (z − z2) (z − z1),
Z = (z − F) (1 − z2) − (G2− 2GG′z + G′2) / 2A2n2
= (z − D) (1 − z2) − (G − G′z)2/ 2A2n2
= (z − E) (1 − z2) − (G′ − Gz)2/ 2A2n2
= (z3− z) (z − z2) (z − z1),
(2)
1 > z3> z > z2> −1, D, E > z1,
(3)
z1+ z2+ z3= F = D − G′2/ 2A2n2= E − G2/ 2A2n2,
(4)
and expressed by the inverse elliptic function
(5)
z = z2sn2mt + z3cn2mt, κ2= (z3− z2) / (z3− z1).
(6)
Equation (25) and (29) § 8 is changed to
(7)
(8)
while ψ and ῶ change places in (26).
The Jacobian elliptic parameter of the third elliptic integral in (7) can be given by ν, where
(9)
where f is a real fraction,
(10)
(11)
with respect to the comodulus κ′.
Then, with z = E, and
2ZE= −{ (G′ − GE) / An}2,
(12)
if II denotes the apsidal angle of ῶ, and T the time of a single beat of the axle, up or down,
= ½πf + Kznf K′,
(13)
in accordance with the theory of the complete elliptic integral of the third kind.
Interpreted geometrically on the deformablehyperboloid, flattened in the plane of the focal ellipse, if OQ is the perpendicular from the centre on the tangent HP, AOQ = amfK′, and the eccentric angle of P, measured from the minor axis, is am(1 − f) K′, the eccentricity of the focal ellipse being the comodulus κ′.
A point L is taken in QP such that
QL/OA = znfK′,
(14)
QV, QT, QP = OA (zs, zc, zd) fK′;
(15)
and with
mT = K, m/n = √ (z3− z1) /2 = OA/k,
(16)
(17)
(18)
By choosing for f a simple rational fraction, such as ½,1⁄3, ¼,1⁄5, ... an algebraical case of motion can be constructed (Annals of Mathematics, 1904).
Thus with G′ − GE = 0, we have E = z1or z2, never z3; f = 0 or 1; and P is at A or B on the focal ellipse; and then
ῶ = −pt, p = G/2A,
(19)
(20)
sin θ exp (ψ + pt) i = i√ [(−z2− z3) (z − z1)] + √ [(z3− z) (z − z2)],
(21)
sin θ exp (ψ + pt)i = i√ [(−z1− z3) (z − z2)] + √ [(z3− z) (z − z1)],
(22)
Thus z2= 0 in (22) makes G′ = 0; so that if the stalk is held out horizontally and projected with angular velocity 2p about the vertical axis OC without giving any spin to the wheel, the resulting motion of the stalk is like that of a spherical pendulum, and given by
= i sin α √ (sec α cos θ) + √ [(sec α + cos θ) (cos α − cos θ)],
(23)
if the axis falls in the lowest position to an angle α with the downward vertical.
With z3= 0 in (21) and z2= −cos β, and changing to the upward vertical measurement, the motion is given by
sin θ eψi= eint√ ½ cos β [√ (1 − cos β cos θ) + i√ (cos β cos θ − cos2θ)],
(24)
and the axis rises from the horizontal position to a series of cusps; and the mean precessional motion is the same as in steady motion with the same rotation and the axis horizontal.
The special case of f = ½ may be stated here; it is found that
(25)
ρ2= a2(κ − x2),
(26)
(27)
L = ½ (1 − κ) + λp/n,
(28)
so that p = 0 and the motion is made algebraical by taking L = ½ (1 − κ).
The stereoscopic diagram of fig. 12 drawn by T. I. Dewar shows these curves for κ =15⁄17,3⁄5, and1⁄3(cusps).
10. So far the motion of the axis OC’ of the top has alone been considered; for the specification of any point of the body, Euler’s third angle φ must be introduced, representing the angular displacement of the wheel with respect to the stalk. This is given by
(1)
(2)
It will simplify the formulas by cancelling a secular term if we make C = A, and the top is then called aspherical top; OH becomes the axis of instantaneous angular velocity, as well as of resultant angular momentum.
When this secular term is restored in the general case, the axis OI of angular velocity is obtained by producing Q′H to I, making
(3)
and then the four vector components OC′, C′K, KH, HI give a resultant vector OI, representing the angular velocity ω, such that
OI/Q′I = ω/R.
(4)
The point I is then fixed on the generating line Q′H of the deformable hyperboloid, and the other generator through I will cut the fixed generator OC of the opposite system in a fixed point O′, such that IO′ is of constant length, and may be joined up by a link, which constrains I to move on a sphere.
In the spherical top then,
(5)
depending on the two elliptic integrals of the third kind, with pole at z = ±1; and measuring θ from the downward vertical, their elliptic parameters are:—
(6)
(7)
(8)
(9)
Then if v′ = K + (1 − f′)K′i is the parameter corresponding to z = D, we find
f = f2− f1, f′ = f2+ f1,
(10)
v = v1+ v2, v′ = v1− v2.
(11)
The most symmetrical treatment of the motion of any point fixed in the top will be found in Klein and Sommerfeld, Theorie des Kreisels, to which the reader is referred for details; four new functions, α, β, γ, δ, are introduced, defined in terms of Euler’s angles, θ, ψ, φ, by
α = cos ½θ exp ½ (φ + ψ) i,
(12)
β = i sin ½θ exp ½ (−φ + ψ) i,
(13)
γ = i sin ½θ exp ½ (φ − ψ) i,
(14)
δ = cos ½θ exp ½ (−φ − ψ) i.
(15)
Next Klein takes two functions or co-ordinates λ and Λ, defined by
(16)
and Λ the same function of X, Y, Z, so that λ, Λ play the part of stereographic representations of the same point (x, y, z) or (X, Y, Z) on a sphere of radius r, with respect to poles in which the sphere is intersected by Oz and OZ.
These new functions are shown to be connected by the bilinear relation
(17)
in accordance with the annexed scheme of transformation of co-ordinates—
where
ξ = x + yi, η = −x + yi, ζ = −z,Ξ = X + Yi, Η = −X + Yi, Ζ = −Z;
(18)
and thus the motion in space of any point fixed in the body defined by Λ is determined completely by means of α, β, γ, δ; and in the case of the symmetrical top these functions are elliptic transcendants, to which Klein has given the name ofmultiplicative elliptic functions; and
αδ = cos2½θ, βγ = −sin2½θ,αδ − βγ = 1, αδ + βγ = cos θ,√ ( −4αβγδ) = sin θ;
(19)
while, for the motion of a point on the axis, putting Λ = 0, or ∞,
λ = β/δ = i tan ½θeψi, or λ = α/γ = −i cot ½θeψi,
(20)
and
αβ = ½i sin θeψi, αγ = ½i sin θeψi,
(21)
giving orthogonal projections on the planes GKH, CHK; and
(22)
the vectorial equation in the plane GKH of the herpolhode of H for a spherical top.
When f1and f2in (9) are rational fractions, these multiplicative elliptic functions can be replaced by algebraical functions, qualified by factors which are exponential functions of the time t; a series of quasi-algebraical cases of motion can thus be constructed, which become purely algebraical when the exponential factors are cancelled by a suitable arrangement of the constants.
Thus, for example, with f = 0, f′ = 1, f1= ½, f2= ½, as in (24) § 9, where P and P′ are at A and B on the focal ellipse, we have for the spherical top
(1 + cos θ) exp (φ + ψ − qt) i= √ (sec β − cos θ) √ (cos β − cos θ) + i (√ sec β + √ cos β) √ cos θ,