The general theorem which he established may now be stated. Consider any path joining PQ, and moving with the fluid, so that the line contains always the same fluid particles. Let u̇, v̇, ẇ be the time-rates of change of u, v, w at an element ds of the path, at any instant, and du, dv, dw the excesses of the values of u, v, w at the terminal extremity of ds above the values at the other extremity; then the time-rate of variation of udx + vdy + wdz is u̇dx + v̇dy + ẇdz + udu + vdv + wdw or u̇dx + v̇dy + ẇdz + qdq, where q has the meaning specified above. Thus if S be the flow for the whole path PQ, and Ṡ its time-rate of variation, S' denote the sum of u̇dx + v̇dy + ẇdz along the path from P to Q, and q₁, q₀ the resultant fluid velocities at Q and P, we get Ṡ = S' + ½(q₁² － q₀²). This is Thomson's theorem. If the curve be closed, that is, if P and Q be coincident, q₁ = q₀ and Ṡ = S'. But in certain circumstances S' is zero, and so therefore is also Ṡ. Thus in the circumstances referred to, as the closed path moves with the fluid Ṡ is continually zero, and it follows that if Ṡ is zero at any instant it remains zero ever after. But Ṡ is only zero if u, v, w are derivable from a potential, single valued in the space in which the closed path is drawn, so that the path could be shrunk down to a point without ever passing out of such space. In a perfect fluid if this condition is once fulfilled for a closed curve moving with the fluid, it is fulfilled for this curve ever after. The circumstances in which S' is zero are these:--the external force, per unit mass, acting on the fluid at any point is to be derivable from a potential-function, and the density of the fluid is to be a function of the pressure (also a function of the coordinates); and these functions must be such as to render S' always zero for the closed path. This condition is manifestly fulfilled in many important cases; for example, the forces are derivable from a potential due to actions, such as gravity, the origin of which is external to the fluid; and the density is a function of the pressure (in the present case it is a constant), such that the part of S' which depends on pressure and density vanishes for the circuit.