is negative, and then the axis makes r[V-](-A)/[pi] nutations per second. Otherwise, if A is positive _ / dy rt = | ---------------------- _/ y[root](A + 2By + Cy²) 1 sh^(-1) [root]A[root](A + 2By + Cy²) 1 ch^(-1) A + By = ------- ---------------------------- = ------- ---------------, (26) [root]A ch^(-1) y[root](B² ~ AC) [root]A sh^(-1) y[root](B² ~ AC) 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 ½[theta] = x, (Nxe[psi]i)^3/2 = (x + 1)[root]X1 + i(x - 1)[root]X2, (27) X1 -- = ±ax^4 + 2ax³ ± 3(a + b)x² + 2bx ± b, (28) X2 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). 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 by c1 = W + W´[alpha], c2 = W + W´[beta], (1) where [alpha], [beta] are numerical factors depending on the external shape; and if the C.G. is moving with velocity V at an angle [phi] with the axis, so that the axial and broadside component of velocity is u = V cos [phi], v = V sin [phi], the total momentum F of the medium, represented by the vector OF at an angle [theta] with the axis, will have components, expressed in sec. lb.,