MONATOMIC GAS: DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 141 
In order to make the former definitions of S 0s , S rs applicable to this case, we must 
adopt certain conventions as to the interpretation of (5'28), (5'29), to allow for negative 
quantities. These conventions are (a) that in r C m>s C n the positive numerical values 
of r and s, m and n* are to be used in all cases : and ( b ) that r—m and s — n retain 
the same signs as r and s respectively even when m — r, n = s. The latter rule 
preserves, in relation to f rs> the distinction between +0 and —0 which in certain 
cases we desire to maintain. 
CD 
(e) A Symmetrical Expression for 2 (a r — a_ r ). 
0 
In discussing 2(a r —a_ r ) it is convenient to change. our notation for V r (a mn )’ 
writing it in the form V ( r a mn ) to denote the determinant whose (m„ n) th element is 
r a mn . If we define r a mn by the equations 
(5'31) r a mn = a mn (m r), r a„ = 1, r a m0 = r a m ,_ 0 
we make V ( r a mn ) identical with V r (a mn ) as defined in § 5 ( b ), and therefore, by (5'21), 
(5'32) 
a r 
V (rO 
V (O ’ 
f 0 ). 
By applying the operation of continued differencing by rows to V ( r a mn ) we transform 
it into V (§ 0n _ r a mn ), where § 0n . r a mn is defined by (5'28) (putting f mn = r a mn , and making 
no distinction between f m0 and f m% _ 0 — cf. the third equation of (5*31)). Now it is 
readily evident that 
(5*33) $ 0 n. T a mn = r), 4..r®™ = 0 ( n ^ 0), 4. r « r0 = 1, 
so that V (S 0n . r a mn ), and consequently, also, V r (a mn ), is equal to V' r (S 0n- a mn ), this being 
defined as identical with V (§ 0n _a mn ) except that in the r th column the central element 
is unity, while all the others are zero. Hence V' r (S 0n a mn ) is clearly equal to the minor 
of the r th element of the central row in v (<LO- If we replace the elements of this 
central row by +1 (mf>0) or — l(m<0), and denote the result by V' (S 0n a mn ), we 
may evidently write (cf. (5’32)) 
(5'34) 
_ v ' (3 0 .a,„) _ V' (<?.),«„») 
v («»,) v 
by (5-30). 
We next apply to V (S 0n a mn ) and V' (S 0n a mn ) the operation of continued differencing 
by columns, so as to transform them into V ($ mn a mn ) and V' (S mn a mn ). We make one 
slight difference of rule here, as compared with the former differencing by rows : that 
is, in (5'29) we shall preserve the distinction between m — 0 and m — — 0, writing f n 
* The signs of m, n are the same, of course, as those of r, s respectively. 
x 2 
