THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC CAS. 307 
The Integration with respect to p. 
§7 (F) On referring back to equations (81) to (88) it is clear that before executing 
the integration with respect to p, in the calculation of h (rs ) and c (rs), we must 
subtract from I (rs, xu) and J (rs, xis), as given by (107) to (l 10), their values 
corresponding to y 13 = 0. Now when X 12 = 0- we have 
1 - cos xia = 0, P* (cos X 12 ) = k 
Thus where P,. (cos xv>) occurs alone in (107) to (llO), it must be replaced by 
P 7 , (c-os'xis) - 1 in the expressions for L (rs) and c(rs), the terms P /c (cos X 12 ) (l— cos X 12 ) 
and P ft (cos X 12 ) (1 — cos X 12) 2 remaining- unchanged, since the corresponding terms in 
I (rs, 0) and J (rs, 0) vanish. 
The variable p is involved in b (rs) and c (rs) only through p dp and Xi 2 > the lat ter 
being also a function of C R . We may therefore formally execute the integration 
with respect to p by writing 
(111) 0*12 (C K ) = (2&+ 1) Wi) - ' 4 C E f {1 -P, (cos X 12 )} P dp, 
Jo 
„co 
(112) ( p'\ 2 (C B ) = (2 k + 1) (/n v 2 y k C H I (1 - cos X 12 ) Pa (cos X i 2 ) p dp, 
Jo 
~CD 
(113) <p"\ 2 (C H ) = (2&+ 1) (mn 2 y ,k I (l -cos Xl 2 ) 2 Pa (cos X i 2 ) V dp. 
Jo 
The nature of these functions depends on the law of inter-action between molecules 
at collisions, and by keeping this law unspecified we retain the utmost generality 
in our theory, which implies no property of the molecules save that of spherical 
symmetry. 
By means of the well-known equation 
(114) (&+1) Pjt+i (cos x )-(2^+1) cos x Pa(cos x ) + &P /c _i(cosx) = 0 
the function (p' k vl (C R ) can be expressed in terms of 0* 12 (C K ), for different values of k, 
as follows :— 
(115) 
<P'\ 2 (Ok) = 
k+1 
2k + 3 
I k+ \oAC ll )- ( f>\ 2 (G R ) + 
k 
2k-1 
0*7*12 (C B ), 
and by a repeated application of (114) we may obtain a similar expression (involving 
<P\ 2 (C H ) for l = k, k± 1, k± 2) for (C B ). 
To avoid unnecessary formulae, we shall not write down the forms taken by b (rs) 
and c {rs) on substitution of the results of this section till after we have considered 
the next step in the integration. 
