(4)where X, Y, Z, L, M, N denote components of external applied force on the body.These equations are proved by taking a line fixed in space, whose direction cosines are l, m, n, thendl= mR − nQ,dm= nP − lR,dn= lQ − mP.dtdtdt(5)If P denotes the resultant linear impulse or momentum in this directionP = lx1+ mx2+ nx3,(6)dP=dlx1+dmx2+dnx3dtdtdtdt+ ldx1+ mdx2+ ndx3,dtdtdt= l(dx1− x2R + x3Q)dt+ m(dx2− x3P + x1R)dt+ n(dx3− x1Q + x2P)dt= lX + mY + nZ,(7)for all values of l, m, n.Next, taking a fixed origin Ω and axes parallel to Ox, Oy, Oz through O, and denoting by x, y, z the coordinates of O, and by G the component angular momentum about Ω in the direction (l, m, n)G = l (y1− x2z + x3y)+ m (y2− x3x + x1z)+ n (y3− x1y + x2x).(8)Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin O, so thatx = y = z = 0, butdx= U,dy= V,dz= W,dtdtdtdG= l(dy1− y2R + y3Q − x2W + x3V)dtdt+ m(dy2− y3P + y1R − x3U + x1W)dt+ n(dy3− y1Q + y2P − x1V + x2U)dt= lL + mM + nN,(9)for all values of l, m, n.When no external force acts, the case which we shall consider, there are three integrals of the equations of motion(i.) T = constant,(ii.) x12+ x22+ x32= F2, a constant,(iii.) x1y1+ x2y2+ x3y3= n = GF, a constant;and the dynamical equations in (3) express the fact that x1, x2, x3are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y1, y2, y3moves as if subject to a couple of componentsx2W − x3V, x3U − x1W, x1V − x2U,(10)and the resultant couple is therefore perpendicular to F, the resultant of x1, x2, x3, so that the component along OF is constant, as expressed by (iii).If a fourth integral is obtainable, the solution is reducible to a quadrature, but this is not possible except in a limited series of cases, investigated by H. Weber, F. Kötter, R. Liouville, Caspary, Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian mathematicians; and the general solution requires the double-theta hyperelliptic function.49. In the motion which can be solved by the elliptic function, the most general expression of the kinetic energy was shown by A. Clebsch to take the formT = ½p (x12+ x22) + ½p′x32+ q (x1y1+ x2y2) + q′x3y3+ ½r (y12+ y22) + ½r′y32(1)so that a fourth integral is given bydy3/ dt = 0, y3= constant;(2)dx3= x1(qx2+ ry2) − x2(qx1+ ry1) = r (x1y2− x2y1),dt(3)1(dx3)2= (x12+ x22) (y12+ y22) − (x1y1+ x2y2)2r2dt= (x12+ x22) (y12+ y22) − (FG − x3y3)2= (x12+ x22) (y12+ y22+ y32− G2) − (Gx3− Fy3)2,(4)in whichx12+ x22= F2− x32, x1y1+ x2y2= FG − x3y3,(5)r (y12+ y22) = 2T − p(x12+ x22) − p′x32− 2q (x1y1+ x2y2) − 2q′x3y3− r′y32= (p − p′) x32+ 2 (q − q′) x3y3+ m1,(6)m1− 2T − pF2− 2qFG − r1y32(7)so that1(dx3)2= X3r2dt(8)where X3is a quartic function of x3, and thus t is given by an ellipticintegral of the first kind; and by inversion x3is in elliptic function of the time t. Now(x1− x2i) (y1+ y2i) = x1y1+ x2y2+ i (x1y2− x2y1) = FG − xy3y3+ i √ X3,(9)y1+ y2i=FG − x3y3+ i √ X3,x1+ x2ix12+ x22(10)d(x1+ x2i) = −i [ (q′ − q) x3+ r′y3] + irx3(y1+ y2i),dt(11)dlog (x1+ x2i) = −(q′ − q) x3− r′y3+ rx3FG − x3y3+ i √ X3,dtiF2− x32(12)dlog√x1+ x2i= −(q′ − q) x3− (r′ − r) y3− FrFy3− Gx3,dtix1− x2iF2− x32(13)requiring the elliptic integral of the third kind; thence the expression of x1+ x2i and y1+ y2i.Introducing Euler’s angles θ, φ, ψ,x1= F sin θ sin φ, x2= F sin θ cos φ,x1+ x2i = iF sin θε−ψi, x3= F cos θ;(14)sin θdψ= P sin φ + Q cos φ,dt(15)F sin2θdψ=dTx1+dTx2dtdy1dy2= (qx1+ ry1) x1+ (qx2+ ry2) x2= q (x12+ x22) + r (x1y1+ x2y2)= gF2sin2θ + r (FG − x3y3),(16)ψ − qFt =∫FG − x3y3Fr dx3,F2− x32√ X3(17)elliptic integrals of the third kind.Employing G. Kirchhoff’s expressions for X, Y, Z, the coordinates of the centre of the body,FX = y1cosxY+ y2cosyY+ y3coszY,(18)FY = −y1cosxX+ y2cosyX+ y3coszX,(19)G = y1cosxZ+ y2cosyZ+ y3coszZ,(20)F2(X2+ Y2) = y12+ y22+ y32− G2,(21)F(X + Yi) =Fy3− Gx3+ i √ X3εψi.√ (F2− x32)(22)Suppose x3 − F is a repeated factor of X3, then y3= G, andX3= (x3− F)2[p′ − p(x3+ F)2+ 2q′ − qG (x3+ F) − G2],rr(23)and putting x3− F = y,(dy)2= r2y2[4p′ − pF2+ 4q′ − qFG − G2+ 2(2p′ − pF +q′ − qG)y +p′ − py2],dtrrrrr(24)so that the stability of this axial movement is secured ifA = 4p′ − pF2+ 4q′ − qFG − G2rr(25)is negative, and then the axis makes r√(-A)/π nutations per second. Otherwise, if A is positivert =∫dyy √ (A + 2By + Cy2)=1sh−1√ A √ (A + 2By + Cy2)=1ch−1A + By,√ Ach−1y√ (B2~ AC)√Ash−1y √ (B2~ AC)(26)and the axis falls away ultimately from its original direction.A number of cases are worked out in the American Journal of Mathematics (1907), in which the motion is made algebraical by the use of the pseudo-elliptic integral. To give a simple instance, changing to the stereographic projection by putting tan ½θ = x,(Nx eψi)3/2= (x + 1) √ X1+ i (x − 1) √ X2,(27)X1= ± ax4+ 2ax3± 3 (a + b) x2+ 2bx ± b,X2(28)N3= −8 (a + b),(29)will give a possible state of motion of the axis of the body; and the motion of the centre may then be inferred from (22).
(4)
where X, Y, Z, L, M, N denote components of external applied force on the body.
These equations are proved by taking a line fixed in space, whose direction cosines are l, m, n, then
(5)
If P denotes the resultant linear impulse or momentum in this direction
P = lx1+ mx2+ nx3,
(6)
= lX + mY + nZ,
(7)
for all values of l, m, n.
Next, taking a fixed origin Ω and axes parallel to Ox, Oy, Oz through O, and denoting by x, y, z the coordinates of O, and by G the component angular momentum about Ω in the direction (l, m, n)
G = l (y1− x2z + x3y)+ m (y2− x3x + x1z)+ n (y3− x1y + x2x).
G = l (y1− x2z + x3y)
+ m (y2− x3x + x1z)
+ n (y3− x1y + x2x).
(8)
Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin O, so that
= lL + mM + nN,
(9)
for all values of l, m, n.
When no external force acts, the case which we shall consider, there are three integrals of the equations of motion
(i.) T = constant,(ii.) x12+ x22+ x32= F2, a constant,(iii.) x1y1+ x2y2+ x3y3= n = GF, a constant;
(i.) T = constant,
(ii.) x12+ x22+ x32= F2, a constant,
(iii.) x1y1+ x2y2+ x3y3= n = GF, a constant;
and the dynamical equations in (3) express the fact that x1, x2, x3are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y1, y2, y3moves as if subject to a couple of components
x2W − x3V, x3U − x1W, x1V − x2U,
(10)
and the resultant couple is therefore perpendicular to F, the resultant of x1, x2, x3, so that the component along OF is constant, as expressed by (iii).
If a fourth integral is obtainable, the solution is reducible to a quadrature, but this is not possible except in a limited series of cases, investigated by H. Weber, F. Kötter, R. Liouville, Caspary, Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian mathematicians; and the general solution requires the double-theta hyperelliptic function.
49. In the motion which can be solved by the elliptic function, the most general expression of the kinetic energy was shown by A. Clebsch to take the form
T = ½p (x12+ x22) + ½p′x32+ q (x1y1+ x2y2) + q′x3y3+ ½r (y12+ y22) + ½r′y32
T = ½p (x12+ x22) + ½p′x32
+ q (x1y1+ x2y2) + q′x3y3
+ ½r (y12+ y22) + ½r′y32
(1)
so that a fourth integral is given by
dy3/ dt = 0, y3= constant;
(2)
(3)
= (x12+ x22) (y12+ y22) − (FG − x3y3)2= (x12+ x22) (y12+ y22+ y32− G2) − (Gx3− Fy3)2,
= (x12+ x22) (y12+ y22) − (FG − x3y3)2
= (x12+ x22) (y12+ y22+ y32− G2) − (Gx3− Fy3)2,
(4)
in which
x12+ x22= F2− x32, x1y1+ x2y2= FG − x3y3,
(5)
r (y12+ y22) = 2T − p(x12+ x22) − p′x32− 2q (x1y1+ x2y2) − 2q′x3y3− r′y32= (p − p′) x32+ 2 (q − q′) x3y3+ m1,
r (y12+ y22) = 2T − p(x12+ x22) − p′x32
− 2q (x1y1+ x2y2) − 2q′x3y3− r′y32
= (p − p′) x32+ 2 (q − q′) x3y3+ m1,
(6)
m1− 2T − pF2− 2qFG − r1y32
(7)
so that
(8)
where X3is a quartic function of x3, and thus t is given by an ellipticintegral of the first kind; and by inversion x3is in elliptic function of the time t. Now
(x1− x2i) (y1+ y2i) = x1y1+ x2y2+ i (x1y2− x2y1) = FG − xy3y3+ i √ X3,
(9)
(10)
(11)
(12)
(13)
requiring the elliptic integral of the third kind; thence the expression of x1+ x2i and y1+ y2i.
Introducing Euler’s angles θ, φ, ψ,
x1= F sin θ sin φ, x2= F sin θ cos φ,x1+ x2i = iF sin θε−ψi, x3= F cos θ;
x1= F sin θ sin φ, x2= F sin θ cos φ,
x1+ x2i = iF sin θε−ψi, x3= F cos θ;
(14)
(15)
= (qx1+ ry1) x1+ (qx2+ ry2) x2= q (x12+ x22) + r (x1y1+ x2y2)= gF2sin2θ + r (FG − x3y3),
= (qx1+ ry1) x1+ (qx2+ ry2) x2
= q (x12+ x22) + r (x1y1+ x2y2)
= gF2sin2θ + r (FG − x3y3),
(16)
(17)
elliptic integrals of the third kind.
Employing G. Kirchhoff’s expressions for X, Y, Z, the coordinates of the centre of the body,
FX = y1cosxY+ y2cosyY+ y3coszY,
(18)
FY = −y1cosxX+ y2cosyX+ y3coszX,
(19)
G = y1cosxZ+ y2cosyZ+ y3coszZ,
(20)
F2(X2+ Y2) = y12+ y22+ y32− G2,
(21)
(22)
Suppose x3 − F is a repeated factor of X3, then y3= G, and
(23)
and putting x3− F = y,
(24)
so that the stability of this axial movement is secured if
(25)
is negative, and then the axis makes r√(-A)/π nutations per second. Otherwise, if A is positive
(26)
and the axis falls away ultimately from its original direction.
A number of cases are worked out in the American Journal of Mathematics (1907), in which the motion is made algebraical by the use of the pseudo-elliptic integral. To give a simple instance, changing to the stereographic projection by putting tan ½θ = x,
(Nx eψi)3/2= (x + 1) √ X1+ i (x − 1) √ X2,
(27)
(28)
N3= −8 (a + b),
(29)
will give a possible state of motion of the axis of the body; and the motion of the centre may then be inferred from (22).