354 Prof. L. Natanson on 



(12), and (13), take the place occupied in the ordinary 

 equations by the components <?, /, g, a, b, c, <w, and play 

 precisely the same part. 



§ 9. The constants X and /a, defined by equations (15) and 

 fl4) of the preceding paragraph, are the two coefficients of 

 viscosity in our theory. Authors who have dealt with the 

 problem of viscosity have generally used two constants which 

 they have often denoted by the same symbols : X, jju. Poisson, 

 in the memoir already quoted *, introduces two constants 

 which, at least in the general case, are independent of each 

 other. In 1843, Barre de Saint- Venant t, and in 1845, with 

 clearness and precision, Sir Gr. Gr. Stokes J, pointed out the 

 considerations which led to the conclusion that 



*=-*» a) 



This is the relation arrived at in 1867 by Clerk-Maxwell §, 

 who was guided by the kinetic theory. We find the same 

 relation adopted by KirchhofT ||, Basset H, Lamb **, and many 

 other writers; nevertheless, in 1874, 0. E. Meyer ft proposed 

 a totally different relation. Lastly, Voigf J J has quite recently 

 called into question the existence of any relation whatever 

 connecting the two constants of viscosity. 



One may hope to find some indications regarding the value 

 of the ratio \//jl by calculating, for a fluid, the quantity 

 (already considered by Stokes and by Helmholtz) which has, 

 after Lord Rayleigh, been called the dissipation function. 

 This is the method which was suggested, apparently for the 

 first time, by Jacobi in the theory of elasticity ; it has been 

 followed by Duhem§§ in the theory of viscosity of fluids. 

 In this latter case there undoubtedly exists a dissipation 

 function, and it is always positive. This condition is easily 



* Journal de V.Ecole Toly technique, 20 Cahier, tome xiii. (1831). 



t Comptes JRendus, tome xvii. p. 1240 (1843). 



X Transactions of the Cambridge Philosophical Society, vol. viii. 

 p. 287 (1845) ; ' Mathematical and Physical Papers,' vol. i. p. 75 (1880) ; 

 see §§ 3, 4, & 18. 



§ Philosophical Transactions, vol. clvii. pp. 81-82 (1867). Scientific 

 Papers, vol. ii. p. 69 (1890). 



|| Vorlesungen ilber die Theorie der Warme, p. 193 (1894). 



IT ' A Treatise on Hydrodynamics,' vol. ii. p. 242 (1888). 



** ' Hydrodynamics,' p. 512 (1895). 



tt Crelle's Journal f. reine u. angeiv. Mathematik, Bd. lxxviii. p. 130 

 (1874). Kinetische Theorie der Gase, II. Auflage, Mathem. Zusatze, 

 pp. 112-114 (1899). 



XX Kompendium d. theoretischen Physik, Bd. i. p. 462 (1895). 



§§ Theorie thermodynamigue de la viscosite, du froltement et des faux 

 equilibres chimiques, Paris 1896, p. 52. 



