Гидродинамика

184 ON THE MOTION OF A RIGID BODY First is given a detailled treatment of the case of an ellip ­ soidal cavity ( § 12), When a body containing such a cavity is in rotary motion the lines of flow in the relative motion of the liquid within the cavity represent a family of ellipses formed by intersections by planes conjugate to the direction of the axis of rotation, of the family of concentric ellipsoids similar to the cavity. Such motion is termed by Prof. Joukowsky an elliptic rotation. Next is discussed the problem of cylindrical and of prismatic cavities (§ 13). Taking first the case of rotation about an axis parallel to the generators, in his analysis of the motion of the liquid the author makes use of de Saint-Venant's method (§ 14): i. e. assumes the velocity potential and selects the parameters so as to obtain a simple shape of the cross-section of the cylin ­ der. This method is used in t he investigation of cavities in the shape of an eUiptic cylinder ( § 15, Fig. 6) and of a triangular equilateral prism (§ 15, Fig. 7). The same method is applied in the solution of the problem of a cylindrical cavity, of which the cross-section at right angles to the generators is bounded by two confocal ellipses. As a particular case is considered an elliptic cylinder with a rectan ­ gular partition (Fig- 8) intersecting t he upper and the lower bases of the cylinder along straight lines connecting the foci of the cylinder ( § 16). Lastly (§ 17) is considered a cylindrical cavity having a cross-section in the shape of a sector of a circle. Tur ­ ning to the case of rotation of t h e body about an axis a t right angles t o t he generators ( § 18) the author states Stokes's pro ­ blem of a cavity in the shape of a rectangular parallelepiped { § 19) and gives a solution, by means of Bessel's functions (formula 40), of a new problem of a cavity in the shape of a right circular cylinder (§ 20). Ne3ct is taken up the question of cavities having the shape of solids of revolution, the body itself rotating about an axis at right angles to the axis of the cavity. Prof. Joukowsky shows that the problems relating to this case are solved with the aid of associated spherical functions of the first kind (§ 21, formula 56). In bis analysis he makes use both of the direct method and of de Saint-Venant's method. The last-named-