d FG - x3y3 + i[root]X3 --- log (x1 + x2i) = dti -(q´ - q)x - r´y + rx ---------------------, (12) dti F² - x3² d /x1 + x2i Fy3 - Gx3 --- log \ / -------- = -(q´ - q)x3 - (r´ - r)y3 - Fr---------, (13) dti \/ x1 - x2i F² - x3² requiring the elliptic integral of the third kind; thence the expression of x1 + x2i and y1 + y2i. Introducing Euler's angles [theta], [phi], [psi], x1 = F sin [theta] sin [phi], x2 = F sin [theta] cos [phi], x1 + x2i = iF sin [theta][epsilon]^(-[psi]i), x3 = F cos [theta]; (14) d[psi] sin [theta] ------ = P sin [phi] + Q cos[phi], (15) dt d[psi] dT dT F sin²[theta] ------ = --- x1 + --- x2 dt dy1 dy2 = (qx1 + ry1)x1 + (qx2 + ry2)x2 = qx1² + x2²) + r (x1y1 + x2y2) = gF² sin² [theta] + r(FG - x3y3), (16) _ / FG - x3y3 Fr dx3 [psi] - qFt = | --------- --------, (17) _/ F² - x3² [root]X3 elliptic integrals of the third kind. Employing G. Kirchhoff's expressions for X, Y, Z, the coordinates of the centre of the body, __ __ __ FX = y1 cos xY + y2 cos yY + y3 cos zY, (18) __ __ __ FY = -y1 cos xX + y2 cos yX + y3 cos zX, (19) __ __ __ G = y1 cos xZ + y2 cos yZ + y3 cos zZ, (20) F²(X² + Y²) = y1² + y2² + y3² - G², (21) Fy3 - Gx3 + i[root]X3 F(X + Yi) = --------------------- [epsilon]^[psi]_i. (22) [root](F² - x3²) Suppose x3 - F is a repeated factor of X3, then y3 = G, and _ _ | p´ - p q´ - q | X3 = (x3 - F)² | ------(x3 + F)² + 2------G(x3 + F) - G² |, (23) |_ r r _| and putting x3 - F = y, _ / dy \² | p´ - p q´ - q ( -- ) = r²y² | 4 ------ F² + 4 ------ FG - G² \ dt / |_ r r _ / p´ - p q´ - q \ p´ - p | + 2 ( 2 ------ F + ------ G ) y + ------ y² |, (24) \ r r / r _| so that the stability of this axial movement is secured if p´ - p q´ - q A = 4 ------F² + 4 ------FG - G² (25) r r