In Chapter VI reference has been made to the "Notes on Hydrodynamics" published by Thomson in the _Cambridge and Dublin Mathematical Journal_ for 1848 and 1849. These Notes were not intended to be entirely original, but were composed for the use of students, like Airy's Tracts of fifteen years before. The first Note dealt with the equation of continuity, that is to say, the mathematical expression of the obvious fact that if any region of space in a moving fluid be considered, the excess of rate of flow into the space across the bounding surface, above the rate of flow out, is equal to the rate of growth of the quantity of fluid within the space. The proof given is that now usually repeated in text-books of hydrodynamics. The second Note discussed the condition fulfilled at the bounding surface of a moving fluid. The chief mathematical result is the equation which expresses the fact, also obvious without analysis, that there is no flow of the fluid across the surface. In other words, the component of the motion of a fluid particle in the immediate neighbourhood of the surface at any instant, taken in the direction perpendicular to the surface, must be equal to the motion of the surface in that direction at the same instant. The third Note, published a year later (February 1849), is of considerable scientific importance. It is entitled, "On the Vis Viva of a Liquid in Motion." What used to be called the "vis viva" of a body is double what is now called the energy of motion, or kinetic energy, of the body. The term liquid is merely a brief expression for a fluid, the mass of which per unit volume is the same throughout, and suffers no variation. The fluid, moreover, is supposed devoid of friction, that is, the relative motions of its parts are unresisted by tangential force between them. The chief theorem proved and discussed may be described as follows. The liquid is supposed to fill the space within a closed envelope, which fulfils the condition of being "simply continuous." The condition will be understood by imagining any two points A, B, within the space, to be joined by two lines ACB, ADB both lying within the space. These two lines will form a circuit ACBDA. If now this circuit, however it may be drawn, can be contracted down to a point, without any part of the circuit passing out of the space, the condition is fulfilled. Clearly the space within the surface of an anchor-ring, or a curtain-ring, would not fulfil this condition, for one part of the circuit might pass from A to B round the ring one way, and the other from A to B the other way. The circuit could not then be contracted towards a point without passing out of the ring.