On the Integrated Velocity Equations of Chemical Reactions. 225 
and %) is unity (the lowest initial concentration), hence we find 
, 1 , 1--- 
.,,-1 ^ = haP-^^ . . (10) 
II Ob-ni) 
or, in terms of the function suggested in § 1, 
- = JcaP-'^ . . (11)- 
^ II (^^j-ni) 
So that, for instance, the integrated equation for a quadri-molecular reaction 
in which one molecule of each reactant takes part, becomes 
f ^ S + f ^ ^ + r ^ . = ^^'' (12) 
(no - %i) (^^3 - '/q ) (7^^ - 71,) {71. - 71,) (U^- 71.) {71, - /?.) 
If a, h, c, d, are the initial concentrations, the substitutions 7i-^a — h, 
n, a = c, 71.^ a = d, \ a = A\ ^ = f, reduce this to the form of integral given 
in the text-books. 
§ 3. The Case of Multiple Poles. 
When the initial concentrations of two or more reactants are equal, or 
when the number of molecules of any reactant exceeds unity, the correspond- 
ing integral could be obtained from a consideration of the appropriate limit 
in equation (11). This, though interesting mathematically, is hardly to be 
recommended as a practical method for chemists. It seems preferable to 
consider separately the different types according to the number of different 
concentrations involved in the integrand. 
(a) Single term denoininator. 
This reduces to the comparatively trivial case of all initial concentra- 
tions being equal. If the reaction is N-molecular the differential ecjuation is 
^-^^ = i«^'-'dS .... (13) 
and the integral may be written 
r::2"i +''"'(1) = """'^^ • • (14) 
or 
W-i [(l-^)'-'^-i] = • ■ (15) 
which is obtainable by direct integration. 
* The vanishing factor for^ — i is, of course, excluded from II. 
