THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 299 
In the integrand of (66), the exponential term and C E (whether occurring as an 
explicit factor or implicitly in xi 2 ) are even functions of X, Y, Z. Hence a term such 
as X 0 a Y 0 i Z 0 c X R fl! Y E e Z I / in KQfi — Qi) (F]. + F 2 ) de will contribute nothing to A 12 Qi unless 
a, b, c, d , e, f and d fortiori a + d, b + e, c+f are all separately even. In view of 
what has been proved above, therefore, it appears that in Q (U, V, W) F (U, V, W) 
only the terms which are even in U, V, W separately contribute anything to A 12 Qi. 
Hence if Q is odd in U, only the part of F which is likewise odd in U need be 
considered, while if it is even in U, only the even part of F need be considered. 
Introduction of F (xis)- 
§ 5 (E) We now make the final transformation of A 12 Qi by adopting polar co-ordi¬ 
nates in place of (X 0 , Y 0 , Z 0 ), (X R , Y I{ , Z E ), as follows :— 
(67) A, a = v a (^)- ,fi (^Jjjje-*~ <w+c -ni.(x. 2 )-r(0)}C!o 2 C* f 7 E dC o( 7C E , 
where 
(68) Ii(xis) = 11 I [Qi(U , 1 ,V , 1 ,W , 1 ) {F^Uj, V 1? W,) + F 2 (U 2 , V 2 , W 2 )} dedcos6 0 dcosO^dfodfa, 
(69) 1,(0)= Qi (U l5 Vi, W 1 ){F a (U 1 ,V 1 ,W 1 ) + F 2 (U 2 ,V 2 ,W 2 )}'dedcos0 o dcos0 E d0o^ E - 
Evidently (cf § 4 (G)) the latter is obtained when X 12 is made zero in F (xi 2 ), since X 12 
is not concerned in the integrations of (68), (69), being a function of p and C R only, 
while when Xi 2 =0 we have Q, (Uh, V'i, W'i) = Qi (Ui, Vi, Wi). Flence, in calculating 
A 12 Qi we shall concern ourselves only with F (x J2 ) until we come to the integration 
with respect to p, C E , C 0 . In so doing we shall, from the outset, omit from F (U, V, W) 
those parts which, in accordance with § 5 (D), contribute nothing to the final result. 
§6. The Form of the Function F(U, V, W). 
The two special forms of Q x which we consider are U/Cj 25 and U 1 C 1 2s ; the only 
parts of F(U, V, W) which are relevant in these cases are respectively the part of 
E 1 + E 2 , which is even in V and W 2 , and Oj + 0 2 ; the notation here used is that of 
§ 2 (E), p. 283. From (26) and (30) we see that AUjC/ 5 involves the space derivatives 
1 ST 
of mean properties of the gas only in the form — —, while AlQCi 25 similarly involves 
F O tXs 
only 2 ~ ^ We deduce from this that O (U, V, W) must certainly include 
cx oy cz 
the term 
