The discussion of the third law of motion is particularly valuable, for, as is well known, attention was therein called to the fact that in the last sentences of the Scholium which Newton appended to his remarks on the third law, the rates of working of the acting and reacting forces between the bodies are equal and opposite. Thus the whole work done in any time by the parts of a system on one another is zero, and the doctrine of conservation of energy is virtually contained in Newton's statement. The only point in which the theory was not complete so far as ordinary dynamical actions are concerned, was in regard to work done against friction, for which, when heat was left out of account, there was no visible equivalent. Newton's statement of the equality of what Thomson and Tait called "activity" and "counter-activity" is, however, perfectly absolute. In the completion of the theory of energy on the side of the conversion of heat into work, Thomson, as we have seen, took a very prominent part. After the introduction of the dynamical laws the most interesting part of this chapter is the elaborate discussion which it contains of the Lagrangian equations of motion, of the principle of Least Action, with the large number of extremely important applications of these theories. The originality and suggestiveness of this part of the book, taken alone, would entitle it to rank with the great classics--the _Mécanique Céleste_, the _Mécanique Analytique_, and the memoirs of Jacobi and Hamilton--all of which were an outcome of the Principia, and from which, with the Principia, the authors of the _Natural Philosophy_ drew their inspiration. It is perhaps the case, as Professor Tait himself suggested, that no one has yet arisen who can bend to the fullest extent the bow which Hamilton fashioned; but when this Ulysses appears it will be found that his strength and skill have been nurtured by the study of the _Natural Philosophy_. Lagrange's equations are now, thanks to the physical reality which the expositions and examples of Thomson and Tait have given to generalised forces, coordinates, and velocities, applied to all kinds of systems which formerly seemed to be outside the range of dynamical treatment. As Maxwell put it, "The credit of breaking up the monopoly of the great masters of the spell, and making all their charms familiar in our ears as household words, belongs in great measure to Thomson and Tait. The two northern wizards were the first who, without compunction or dread, uttered in their mother tongue the true and proper names of those dynamical concepts, which the magicians of old were wont to invoke only by the aid of muttered symbols and inarticulate equations. And now the feeblest among us can repeat the words of power, and take part in dynamical discussions which a few years ago we should have left to our betters."