314 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
the absolute temperature. The elements b rs , c rs and the coefficients ft r , y r then become 
functions of the temperature only. 
The coefficient /3_ a is not determined by the above equations, but is given (cf (75)) by 
(148) 8_ a = - 2 ft/(r+1). 
r = 0 
Properties of the Determinants V (b rs ), V (c rs ). 
§ 8 (C) On inspection of (143) and (144) it is evident that 
(149) b„ = b sr , c rs = c sr , 
so that V (b rs ) and V (c rs ) are symmetrical determinants. 
In expression (143) for b rs , the variables of integration, x and y, are never negative, 
so that (cf. (129), (96)) r A*, s A k and B k (r, s ) are essentially positive (or zero) for all 
integral values of r, s, and Jc ; further, since P* (cos y) never exceeds unity, <p 2k (tij) is 
also always positive. It is evident, therefore, that b rs must be essentially positive 
if this can be proved true of 
(150) B 2 *('r+2,s + l) — 2y 2 B 2k (r+l, s +l) + B 2 *(r +1, s + 2). 
Now 
(151) B 2 *(r + 2, s+l)-rB 2/f (r+l, s+l) = S+1 A 2 * [ r+2 A 2k -y 2r+1 A 2k '] 
= s+ij^kl y 
\x 
2k 
r + 2 
V 
_ *+i^2fc( a 
\X 
V\ 2k A 2 ( r + l) t _! (n + f ) t -2k-i 
t = 2k 
(^+1l)< (t — 2k) 
' ( r + l) t (r + -§-), 
2k (t + -%)t (t + 2li 
x 2ty2(r+2-t) + 2 — t) (t — 2Jc) + t(r+%)}, 
' ( r +2) t (r-{-f) t _ 2 k 2 t^. 2 (r+ 2 -t) '-V \' r 1 1 Jt V 1 ~f 2 ft- 2 k pit ,.2 (r+ 2 -t) 
L. "a (t+i), (t-2k) ! J ,r a (t + i), (t + 2k) ! J 
every term of which is positive. Interchanging r and s in (151), and adding the result 
to the latter, we obtain (150), which, with b rs also, in consequence, is essentially 
positive. 
From (151), moreover, it is clear that the numerical coefficients in (151) or (150) 
increase with r or s, and the .same is readily seen to hold good also in the case of 
B* (r, s). As r or s increases, therefore, the numerical coefficients and the degree 
(in x and y) of the integrand of (143) increase, while if both r and s increase, new 
positive terms are added to the integrand. Hence, provided that the functions 
f k (' ry ) satisfy certain simple conditions,* b rs steadily increases with r or s, and the 
consideration of even a single term of (151) or the integrand of (143) shows that this 
increase is without limit, i.e., b rs tends steadily to infinity with r or s. 
* It is easy to see that the increase with y of <p k (ry) is less rapid than that of y; if <£ 2fc (U/) constant 
or steadily increases, though less rapidly than y, b rs will steadily increase with r or s. But much less 
restrictive conditions might be devised, e.g ., if p k (ry) decreases like y~ l , the above result still holds good. 
