of the corresponding simple pendulum, and, therefore, the time of vibration, least possible. To investigate these circumstances for all axes parallel to a given line, through the centre of gravity, let k be the radius of gyration round this line, we have (§ 198), I=W(k2 + h2); and, therefore, if I be the length of the isochronous simple pendulum, h2 + k2 __ (h−−k)2+2hk (h-k)2 =2k+ h The second term of the last of these forms vanishes when h=k, and is positive for all other values of h. The smallest value of l is, therefore, 2k, and this, the shortest length of the isochronous simple pendulum, is realized when the axis of support is at the distance k from the centre of inertia. To find at what distance h, from the centre of inertia the axis must be fixed to produce a pendulum isochronous with the simple pendulum, of given length 7, we have the quadratic equation h2-hl-k2. For the solution to be possible we have seen that I must be greater than, or at least equal to, 2k. If l=2k, the roots of this equation are equal, k being their common value. For any value of greater than l 2k, the equation has two real roots whose sum is equal to l, and product equal to 2: hence, for any distance from the centre of inertia less than k, another distance greater than k, which is a third proportional to it and k, gives the same time of vibration; and the length of the. simple pendulum corresponding to either case, is equal to the sum of the distances of the two axes from the centre of inertia. This sum is equal to the distance between them if the two axes are in one plane, through the centre of inertia, and on opposite sides of this point; and, therefore, for axes thus placed, and not equidistant from the centre of inertia, if the times of oscillation of the body when successively supported upon them are found to be equal, it may be inferred that the distance between them is equal to the length of the isochronous simple pendulum. As a simple pendulum exists only in theory, this proposition was taken advantage of by Kater for the practical determination of the force of gravity at any station. (h) A uniformly heavy and perfectly flexible cord, placed in the interior of a smooth tube in the form of any plane curve, and subject to no external forces, will exert no pressure on the tube if it have everywhere the same tension, and move with a certain definite velocity. For, as in § 592, the statical pressure due to the curvature of the rope per unit of length is T (where σ is the length of the arc AB in that figure) directed inwards to the centre of curvature. Now, the element σ, whose mass is mo, is moving in a curve whose curvature is with velocity (suppose). The requisite force is momv20 ; and for unit of length mv2 Hence if Tmv2 the theorem is true. If we suppose a portion of the tube to be straight, and the whole to be moving with velocity v parallel to this line, and against the motion of the cord, we shall have the straight part of the cord reduced to rest, and an undulation, of any, but unvarying, form and dimensions, running along it with the linear velocity m Suppose the cord stretched by an appended mass of W pounds, and suppose its length 7 feet and its own mass w pounds. Then T= Wg, Im=w, and the velocity of the undulation is When an incompressible liquid escapes from an orifice, the velocity is the same as would be acquired by falling from the free surface to the level of the orifice. For, as we may neglect (provided the vessel is large compared with the orifice) the kinetic energy of the bulk of the liquid; the kinetic energy of the escaping liquid is due to the loss of potential energy of the whole by the depression of the free surface. Thus the pro position at once. 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