LAG RANGE AND HAMILTON 191 



the first to point out clearly the significance of Newton's second interpretation 

 of his Third Law. I have heard Tait tell the story of the search after this 

 interpretation. "The Conservation of Energy," he said to Thomson one 

 day, " must be in Newton somewhere if we can only find it." They set 

 themselves to re-read carefully the Principia in the original Latin, and ere 

 long discovered the treasure in the finishing sentences of the Scholium 

 to Lex in. 



Considerable portions of the earlier sections of the dynamical chapter 

 are, as they themselves point out, simply paraphrased from Newton. But 

 a marked feature of the discussion is the introduction of a new terminology 

 at once precise and suggestive. Moreover, old words are used with clearly 

 defined meanings, and never used except with these meanings. The 

 Conservation of Energy occupies the first place, and the terms Kinetic 

 Energy and Potential Energy give a new unity to the whole treatment. 

 The Moment of Inertia is defined, in the first instance, in terms of kinetic 

 energy. The conditions of equilibrium are established on their true kinetic 

 basis. The principle of virtual velocities and d'Alembert's principle, which 

 the older writers regarded with such reverence, are shown to be special 

 enunciations of the great laws of energy. Gradually, by almost imperceptible 

 advances along several lines of comparatively simple dynamical reasoning, 

 the way is paved for the entrance into the shrine of Lagrange's generalised 

 coordinates ; and thence into the spacious temple of Hamiltonian Dynamics. 

 At the time the book was being planned Thomson seems not to have studied 

 the more recent developments of Lagrange's dynamical method ; and his own 

 mathematical methods were based largely on those of Fourier. As Tait 

 once epigrammatically put it, " Fourier made Thomson." Tait used to tell 

 how, when the chapter on Dynamical Principles was being sketched, he 

 remarked, " Of course we must bring in Hamilton's dynamics." Thomson 

 having expressed unfamiliarity with Hamilton's theory, Tait rapidly sketched 

 it on a sheet of foolscap. Thomson was enraptured, took the sheet off with 

 him to Glasgow, and in a short time had the sections written out very 

 much as they appeared in the first edition of the Treatise. 



The typical examples chosen to illustrate the power of Lagrange's 

 generalised equations are of great variety and interest. They include the 

 gyroscopic pendulum and the hydrokinetic problems of solids moving through 

 perfect fluids. This latter class of problem had been imagined by Thomson 

 as early as 1858; but it was in the pages of "T and T'" that the 



