CONSTITUTED OF SPHERICALLY SYMMETRICAL MOLECULES. 441 



which we shall denote by f v , f' u . When there are slight inequalities of temperature 

 and mass motion in the gas, we shall supjxmo that 



/=/ u {l+F(U,V,W)}, /=.A{l + F'(U',V',W')}, 



where F and F' are expansihle in the fonn of power series (without the constant 

 terms) in the variables (U, V, W) and (U', V, W) respectively. The coefficients 

 will be small quantities which are functions of the velocity and temperature, and 

 their derivatives, at the point considered. Thus F and F' represent the small 

 disturbance from the normal law of distribution, caused by the slight lack of 

 uniformity in the gas. MAXWELL and BOLTZMAITN* considered that the terms of 

 the first three degrees are sufficient for the adequate n-|nvM-]itatii>n of the disturbed 

 state, and I shall follow them in this assumption. Thus we write 



(22) F (U, V, W) = (2/mt) 1 " (a 1 U+o > V + o,W) + 2Am (i2a n U a +22a 12 UV) 



where the factors (2hm) 113 , (2hm), (2/mi) s/a are added merely for convenience in the 

 integration. We shall have a similar equation for F', in which m, a lt a,, ... are 

 replaced by m', a',, a' 2 , .... . Since, by definition, 



III f( u> v ' w ) du dv dw = l > jff f ( u '> v '> w/ ) du> dv> dd = 1, 

 we have 



l+n + a + 3s = 1 l+a'n + a'aa + a'a = 1, 



or 



(23) an + Wja+ass = 0, a' ' + a' ' ffl + a' '33 = 0. 



* By MAXWELL in his memoir "On Stresses in Rarefied Gases," 'Phil. Trans.,' 1879, or 'Scientific 

 Papers,' vol. ii., p. 681 ; by BOLT/MANN in his ' Vorlesungen iibcr Gastheorie,' vol. i., p. 185. In each case 

 the assumption was made in connection with a Maxwellian gas, but there are good grounds for believing 

 that it is equally valid in general. As neither MAXWKI.I, nor DOI.T/MAXX considered it necessary to give 

 any justification for their procedure, I deliberately followed the same course, more especially as the 

 attempt to m.-ike the step perfectly rigorous would have necessitated the introduction of much mathe- 

 matical analysis which would be out of place here. 



I should also mention that EXSKOG ('Phys. Zeitschrift,' xii., 58, January, 1911) has made an attempt to 

 determine directly the form of the function F(w, r, ), applying methods of integration, similar in many 

 ways to those used in this paper to evaluate AQ, to an equation arrived at by BOI.T/.MAXN (' Vorlesungen,' 

 vol. i., p. 114). From the expression for F (, *, i/-) thus obtained EXSKOG deduces values of the coefficients 

 of viscosity ;'ii<l thermal conduction for a simple gas. 



I am indebted id Prof. LAIJMOK for the reference to EXSKOC'S work, of which I was unaware till after 

 this paper had been communicated to the Koyal Society. Note added Oriobtr, 1911. Some of these 

 stati-ments are modified by the last note on p. 483. 



VOL. CCXI. A. 3 L 



