It is to be clearly understood that the motion of a fluid may be irrotational although the value of S does not vanish for every closed path that can be drawn in it. The fluid may occupy multiply continuous space, and the path may or may not be drawn so that S shall be zero; but what is necessary for irrotational motion within any space is that S should vanish for all paths which are capable of being shrunk down to zero without passing out of that space. S need not vanish for a path which cannot be so shrunk down, but it must, if the condition just stated is fulfilled, have the same value for any two paths, one of which can be made to pass into the other by change of position without ever passing in whole or in part out of the space. The potential is always single valued in fluid filling a singly continuous space such as that within a spherical shell, or between two concentric shells; within a hollow anchor-ring the potential, though it exist, and the motion be irrotational, is not single valued. In the latter case the motion is said to be cyclic, in the former acyclic. A number of consequences are deduced from this theorem; and from these the properties of vortices, which had previously been discovered by von Helmholtz, immediately follow. First take any surface whatever which has for bounding edge a closed curve drawn in the fluid, and draw from any element of this surface, of area dS, a line perpendicular to the surface towards the side chosen as the positive side, and calculate the angular velocity ω, say, of the fluid about that normal from the components of angular velocity determined in the manner explained at p. 164. This Thomson called the rotation of the element. Now take the product ωdS for the surface element. It is easy to see that this is equal to half the circulation round the bounding edge of the element. As the fluid composing the element moves the area dS may change, but the circulation round its edge by Thomson's theorem remains unaltered. Thus ω alters in the inverse ratio of dS, and the line drawn at right angles to the surface at dS, if kept of length proportional to ω, will lengthen or shorten as dS contracts or expands. Now sum the values of ωdS for the finite surface enclosed by the bounding curve. It follows from the fact that ωdS is equal to half the circulation round the edge of dS, that this sum, which is usually denoted by ΣωdS, is equal to half the circulation round the closed curve which forms the edge of the surface. Also as the fluid moves the circulation round the edge remains unaltered, and therefore so does also ΣωdS for the elements enclosed by it. It is important to notice that this sum being determined by the circulation in the bounding curve is the same for all surfaces which have the same boundary.