6 INTRODUCTION 



only 80 per cent of 80, i.e. the quantity 64 per cent, 

 remains ; after the time 3/ (3 hours) 80 per cent of 64, 

 i.e. the quantity 51-2, remains of the sugar; after the 

 time \t (4 hours) 80 per cent of 51-2, i.e. the quantity 

 40- 96, and so forth. We say then that when time 

 increases in an arithmetic series, the quantity of 

 cane-sugar decreases in a geometric series. If the 

 quantity of cane-sugar is called z and the quantity 

 at the beginning of the experiment z (we have in 

 this case put z = 100), then the said law regarding 

 the progress of the inversion of the cane-sugar with 

 time, t, is expressed by means of the formula 



log z — log z = bt. 



For the time ^ = 0, i.e. when the sulphuric acid 

 is added to the solution of cane-sugar we have 

 log z = \og 2, i.e. z = z = 100. 



Now if we translate the said law into a graphical 

 expression, we get the ^-curve as a function of the 

 time t (Fig. 2, the lower curve). This ^-curve is a 

 so-called exponential curve. Even to an eye ac- 

 customed to curves it is rather difficult to distinguish 

 this exponential ^-curve from another curve indicat- 

 ing a regular decrease of the quantity of cane-sugar, 

 2, with increasing time, /. The curve does not tell 

 us very much in its general character ; only if we 

 measure special points on it, and determine cor- 

 responding values of z and t, do we get a real 

 representation of the meaning of the curve. In 

 this case a table giving the comparison of calculated 



