Now let the liquid given at rest in such a space be set in motion by any arbitrarily specified variation of position of the envelope. The liquid within will be set in motion in a manner depending entirely on the motion of the envelope. It is possible to conceive of other motions of the liquid than that taken, which all agree in having the specified motion of the surface. Thomson's theorem asserts that the motion actually taken has less kinetic energy than that of any of the other motions which have the same motion of the bounding surface. The motion produced has the property described by the word "irrotational," that is, the elements of the fluid have no spinning motion--they move without rotation. A small portion of a fluid may describe any path--may go round in a circle, for example--and yet have no rotation. The reader may imagine a ball carried round in a circle, but in such a way that no line in the body ever changes its direction. The body has translation, but no spin. Irrotationality of a fluid is secured, as stated above, when the velocity of each element in any direction is the rate of variation per unit distance in that direction of a certain function of the coordinates, the distances, taken parallel to three lines perpendicular to one another and drawn from a point, which specify the position of the particle. In fact, what is called a velocity-potential exists, similar to the potential described in Chapter IV above, for an electric field. This condition, together with the specified motion of the surface, suffices to determine the motion of the fluid. Two important particular consequences were pointed out by Thomson: (1) that the motion of the fluid at any instant depends solely on the form and motion of the bounding surface, and is therefore independent of the previous motion; and (2) that if the bounding surface be instantaneously brought to rest, the liquid throughout the vessel will also be instantly brought to rest. This theorem was afterwards generalised by Thomson (_Proc. R.S.E._, 1863), and applied to any material system of connected particles set into motion by specified velocities simultaneously and suddenly imposed at selected points of the system. It was already known that the kinetic energy of a system of bodies connected in any manner, and set in motion by impulses applied at specified points, was either a maximum or a minimum, as compared with that for any other motion compatible with these impulses, and with the connections of the system. This was proved by Lagrange in the _Mécanique Analytique_ as a generalisation of a theorem given by Euler for a rigid body set into rotation by an impulse.