MONATOMIC GAS: DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 143 
(g) Symmetrical Expressions for'Zr {v x a r + v 2 a_ r ), 2 (p 1 /3 r + v 2 (i_ T ), Z (t/ 1 y r + v 2 y_ r ). 
1 1 0 
As in § 5 ( e ) we may prove that 
■v„/ , \ _ ^0 (^lAn) 
+ ~ ry / * „ \ » 
where V 0 (S 0n a mn ) is identical with V (S 0n a mn ) except in the central row 1 , the elements 
of which are equal to rv x (r f> 0) or rv 2 (r 0), the positive numerical value of r being 
taken in both cases. On applying the operation of continued differencing by columns 
we obtain the result 
(5-37) 
Zr ’ 
1 v (<WW 
where V 0 (S mn a mn ) differs from V (S mn a mn ) onty in the central row, all the elements 
of which are zero save those on either side the centre (r — 1 and r — — l), which are 
respectively equal to v 1 and v 2 . This follows from the fact that 
S r0 r =0 (r^ 1), S r0 r =1 (r = l). 
Again, from (5‘2l), we may prove in the usual way that 
a _ (<LQ 
Pr v (s 0n b mn ) ’ 
where V r and V are the same, except that in the r th column of V r all the elements are 
zero save those on either side the centre (i.e., n = ± 0) which are unity. If in 
V and V r we add half of each of the rows n — ± 0 to the centre row, and subtract 
this new centre row from the rows n = ± 0, V becomes transformed into V (S 0n b' mn ), 
where V mn = b mn , except when n — ± 0, while 
(5 38) b m0 = \ {b m() — b m _ 0 ) — b m , b m _ 0 = jjr{b m _ 0 b m0 ) — b m , b m — b m + ^ [b m0 + b m _ () ). 
Similarly, V r becomes transformed into V r (S 0n b' mn ), identical with V (S 0n b' mn ), except 
that in the r th column all the elements are zero save the central one, which is unity. 
Consequently we may write 
2 (vf r + vf_ r ) 
Vq ($0nb'mn) 
V (S 0n b' mn ) ’ 
where V 0 is the same as V except that in the central row all the elements on the right 
of the centre (m f> 0) are equal to v u all those to the left (m — 0) are equal to v 2 , 
while the central element is zero. We now “ difference ” by columns, with the result 
•v / P I O \ _ ^0 (dmnb'mn) 
(5‘39) 
