226 
Transactions of the Royal Society of South Africa. 
(b) Two-term denominator. 
The type of integrand in this case is 
1 
(l_A)a (^n~\y 
It is resolved into 
vhere 
, {\-\y 1 {n-xy 
(-l)«-.--C,_.^_3i^ . . (17) 
The C's are the usual binomial coefficients. Write a + /3 = N (the total 
number of molecules involved), and the integral becomes 
1 ^'-Hl) 
, V (-1)-^" ^ .. op 
1 tll^-Un) 
-(-i)^(„-T)-Hr ■ • • • 
This holds for a > 1, /? > 1. If a > 2 the last term in (18) may be 
dropped, and the second summation carried out between the limits 1 and /3. 
One can well imagine that the form of the solution (18) may not appeal 
to those for whom it is intended ; but its generality will make it worth 
while mastering its symbols. It embraces, for instance, all the equations 
discussed by Todd,* which he integrates individually, and for which he 
gives individual curves. 
As an example of the application of (18) consider the equation 
=: Jc(a-xy (h—x:) 
Here a = S, = l,N=:i4. Substitution of these values in (18) gives 
'■^"'^ = (^^^e[-'J'(1)-(«-^)-'^'(1) + 4(«)] • (19) 
And for the equation 
If = Ha—xy {h-xy 
where a = 2, /i? = 2, N = 4, we get 
A^^S^ = -^^-^-^^[^'(l) - ^(^0 -r (/^-1)^>)] . (20) 
Substitution of the equivalents given in § 1 leads to the ordinary algebraical 
solutions ; while the use of the tables in § 5 gives appropriate numerical 
results. 
* Loc. cit. 
