An entirely different proof of this proposition is given subsequently in the paper, and depends on a new and very general theorem, which has been described as containing almost the whole theory of the motion of a fluid. This depends on what Thomson called the flow along any path joining any two points P, Q in the fluid. Let q be the velocity of the fluid at any element of length ds of such a path, and θ be the angle between the direction of ds (taken positive in the sense from P to Q) and the direction of q: q cos θ.ds is the flow along ds. If u, v, w be the components of q at ds, parallel to the axes, and dx, dy, dz be the projections of ds on the axes, udx + vdy + wdz is the same thing as q cos θ.ds. The sum of the values of either of these expressions for all the elements of the path between P and Q is the flow along the path. The statement that u, v, w are the space-rates of variation of a function φ (of x, y, z) parallel to the axes, or that q cos θ is the space-rate of variation of φ along ds, merely means that this sum is the same for whatever path may be drawn from P to Q. This, however, is only the case when the paths are so taken that in each case the value of φ returns after variation along a closed path to the value which it had at the starting point, that is, the closed path must be capable of being contracted to a point without passing out of space occupied by irrotationally moving fluid. Since the flow from P to Q is the same for any two paths which fulfil this condition, the flow from P to Q by any one path and from Q to P by any other must be zero. The flow round such a closed path is not zero if the condition is not fulfilled, and its value was called by Thomson the circulation round the path.