53. _The Motion of a Perforated Solid in Liquid._--In the preceding investigation, the liquid stops dead when the body is brought to rest; and when the body is in motion the surrounding liquid moves in a uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on its surface is the opposite of that required to change the motion of the liquid; this has been expressed by the dynamical equations given above. But if the body is perforated, the liquid can circulate through a hole, in reentrant stream lines linked with the body, even while the body is at rest; and no reaction from the surface can influence this circulation, which may be supposed started in the ideal manner described in § 29, by the application of impulsive pressure across an ideal membrane closing the hole, by means of ideal mechanism connected with the body. The body is held fixed, and the reaction of the mechanism and the resultant of the impulsive pressure on the surface are a measure of the impulse, linear [xi], [eta], [zeta], and angular [lambda], [mu], [nu], required to start the circulation. This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences an additional reaction from the circulation which is the opposite of the force required to change the position in space of the circulation impulse; and these extra forces must be taken into account in the dynamical equations. An article may be consulted in the _Phil. Mag._, April 1893, by G. H. Bryan, in which the analytical equations of motion are deduced of a perforated solid in liquid, from considerations purely hydrodynamical. The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by [xi] the resultant axial linear momentum of the circulation. As the ring is moved from O to O´ in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from [alpha]M´U + [xi] and [beta]M´V along Ox and Oy to [alpha]M´U´ + [xi] and [beta]M´V´ along O´x´ and O´y´, (1) the axis of the ring changing from Ox to O´x´; and U = Q cos [theta], V = Q sin [theta], U´ = Q cos ([theta] - Rt), V´ = Q sin ([theta] - Rt), (2) so that the increase of the components of momentum, X1, Y1, and N1, linear and angular, are X1 = ([alpha]M´U´ + [xi])[cos] Rt - [alpha]M´U - [xi] - [beta]M´V´ sin Rt = ([alpha] - [beta])M´Q sin ([theta] - Rt) sin Rt - [xi] ver Rt (3) Y1 = ([alpha]M´U´ + [xi]) sin Rt + [beta]M´V´ cos Rt - [beta]M´V = ([alpha] - [beta])M´Q cos ([theta] - Rt) sin Rt + [xi] sin RT, (4)