196 



CALCULATION OF DATA 



gives a curve like that in Fig. 14-2. Rotation of the axes yields another 

 expression, xy = k, which is graphically represented in Fig. 14-3. This 

 expression is quite commonly represented in the results of experiments 



Fig. 14-2. ^ - ^ = 1 



a- b- 



Fig. 14-3. xy = k 



because it indicates that y is inversely proportional to x, or y = k/x. Still 

 another transformation involving rotation and translation of the axes 

 yields the curve in Fig. 14-4, with the equation 



Hx -\- Ky — xy =^ 



where K is a constant with a small negative value and H is a constant 

 with a larger positive value. Such a curve might result in an experiment 

 where, for example, an over-all rate is controlled and limited by two sep- 

 arate factors. When x is small, the rate is limited by x, so that the 

 rate increases rapidly as x increases. When x is large, some other factor 

 limits the rate, so further increases in x make little difference in the 

 rate. 



Exponential and logarithmic expressions also occur frequently in the 

 treatment of experimental results. The general exponential equation, 

 x = a" (where commonly a = 2, e, or 10) can be written in the form 

 y = \og„x. This equation produces the curve of Fig. 14-5. Because of 

 its curvature, this graph is difficult to interpret. Notice the similarity 

 to Fig. 14-4. The logarithmic curve is easily converted to a straight line 

 however. If we set z = log„ x, then y = z, which is an equation for a 



