RECENT PROGRESS IN nYDRODYNAMICS. 63 



was Tbomson,' in 1849. The theorem as enunciated by him is, ' If the 

 bounding surfece of a liquid primitively at rest be made to A'ary in a 

 given arbitrary manner, the vis-viva of the entire liquid at each instant 

 will be less than it would be if the liquid had any other motion consistent 

 with the given motion of the bounding surface.' This theorem was 

 afterwards, in Thomson and Tait's 'Natural Philosophy,' extended to 

 dynamical problems in general. From this theorem, he states three 

 corollaries — (1) The condition that udx + vdij + lodz must be a complete 

 differential is, in addition to the kinematical conditions, sufficient to 

 determine the motion. (2) The motion of the fluid at any time is inde- 

 pendent of the preceding motion, and depends solely on the given form 

 and normal motion of the bounding surface at the instant. (3) If the 

 bounding surface be brought to rest, the liquid will at the same instant 

 be reduced to rest. None of the theorems contained in these corollaries 

 were new. But the first corollary forms, I believe, the first definite proof 

 of the uniqueness of the solution obtained when the fluid is irrotational 

 and the normal motion of the surface is given. Former writers, thouo-h 

 convinced of its truth, had not succeeded in arriving at a formal proof 

 of it. In 1843, Stokes''^ writes that it is a recognised principle, that 

 when a problem is determinate, any solution which satisfies all the 

 requisite conditions is the solution of the problem ; and then states that 

 the problem is determinate — a proof based on experiment. The second 

 and third are also proved by Stokes in the same paper, and by Cauchy.^ 

 The theorem that the velocity cannot be a maximum at any point of the 

 fluid was given by Maxwell* in a Senate House paper, and in his 

 lectures at Cambridge, though not stated in the hydrodynamical form, 

 Thomson also gives the now Avell-known application of Green's theorem 

 to the energy of irrotational motion within a boundary. The analogous 

 expressions for the energy, and its variation with the time in the case of 

 rotational motion, were given by Helmholtz in his vortex paper, and for 

 irrotational motion in multiply continuous space by Thomson.-^ 



(h) Vortex Motion. 

 During the last forty years, without doubt, the most important 

 addition to the theory of fluid motion has been in our knowledo-e of 

 the properties of that kind of motion where the velocities cannot be 

 expressed by means of a potential. Certainly, before this we knew some- 

 thing. Stokes' researches had shown the kinematical nature of the 

 motion — the rotation of its small parts '' — and we also knew that it 



' ' Notes on Hydrodynamics,' Caiiib. and BuM. MafJi. Jour. iv. p. 90. 



'■' ' On some cases of fluid motion,' Ciimh. Phil. Trans, viii. 



' ' Mum. sm- la Tlieorie des Ondes,' Mem. dvs Sav. Etranrjtrs (1827). 



■* Sen. Ho., Thursday afternoon, 1873. 



^ 'Vortex motion,' Roy. Soc. Edin. Trans., xxv. 



" ' On the theories of the internal friction of fluids in motion,' kc, Caml. Phil. 

 Trans, viii. 



As to the true nature of this rotation, reference may be made to a discussion 

 between Helmholtz and Bertrand in the Comptes Hendus, Ixvi. and Ixvii (1868). 

 Bertrands objection reduced itself to a question of the use of the word rotation! 

 The nature of the small displacements of a continuous medium are very fully treated 

 in Thomson and Tait's Xatnral Philosophy/. The reader who desires to enter more 

 fully into tlie theory of the above laws will find much to interest him in the follow- 

 ing papers, not mentioned in the text : — 



E. J. Nanson, Messenger of Math., (2), iii. p. 120 and (2) vii. p. 182. 



C. W. Merrifield, Ibid. (2) iv. p. 105 



H. Lamb, Ibid. (2) vii. p, 41, shows the connection between 



Helmholtz"s and Thomson's methods of proofs. 



