208 MESSRS. A. V. HARCOURT AND W. E8S0N ON TllE LAWS OF CONNEXION 



inquire how their constants were related to the number of units of sulphuric acid 

 employed. Some other expression for the course of the reaction had to be sought, and 

 a consideration of the circumstances under which the reaction now takes place furnished 

 the needful clue. 



If any quantity of a substance be placed under conditions under which it gradually 

 undergoes a change, and if both the conditions and the quantity of the substance be by 

 a due system of compensation kept constant, the rate of change will be uniform ; that 

 is to say, the amount of substance which disappears in a unit of time will be always 

 the same. This amount therefore, for any particular unit of time, is a certain fixed 

 proportion of the total quantity undergoing change. If, now, the system of com- 

 pensation be so far disturbed as that while other conditions remain the same the quan- 

 tity of substance be allowed to diminish, it seems probable that the total amount of 

 change occurring at any moment will be proportional to the quantity of substance then 

 remaining. 



Adopting this hypothesis, the law of connexion between the quantity of substance 

 remaining at any time, and the time during which the change has proceeded, may be 

 found in the following way : — 



Let y represent the amount of substance remaining after the change has proceeded 

 for a time x, and let dy be the diminution of the substance during an infinitesimal time 



dx^ then -r- represents the amount of substance which disappears in a unit of time ; and 



this amount is by the hypothesis proportional to the quantity of substance remaining ; so 



that we have the equation 



dy 



which gives 



y=ae-", 



where a is the quantity of substance at the commencement of the change. 



This equation expresses the fact that the quantities of substance remaining after a 

 series of intervals of time increasing in arithmetical progression, form a series in geome- 

 trical progression. After the intervals 0, 1,2,3,... minutes, the quantities of substance 

 remaining are 



a geometric series of which the ratio is e'". The curve which expresses this relation 

 between x and y is a logarithmic curve. 



As the actual determinations of the oxidizing residue had been made at equal inter- 

 vals of time, it was an easy matter to test the applicability of this hypothesis. It 

 appeared that although these numbers were in most cases nearly in geometrical pro- 

 gression through a considerable range, they were not so throughout, and the values of 

 y calculated on this hypothesis differed more from the experimental values than those 

 before obtained from the equation of an oblique hyperbola. A single experiment, 

 however, made at the close of this series, in which no sulphuric acid was taken, showed 



