458 RESEARCH IN PROTOZOOLOGY 



RATE CURVES 



Under the heading of correlation we discussed the method of 

 expressing a relationship between variables in the case where 

 the relationship took on a linear form ; that is, in the case where 

 a change of a unit in one of the variables was accompanied by a 

 constant change in the other variable. There are of course many 

 cases in which the variables under consideration are not thus 

 related, but a general discussion of non-linear correlation is beyond 

 the scope of this article. It will, however, be advisable for us to 

 consider one non-linear case due to its importance in expressing 

 rates of reaction. The case in question is that of geometric in- 

 crease or decrease, or the curve of constant proportional increase 

 as contrasted with that of constant absolute increase treated under 

 the heading of correlation. 



If the two variables under discussion are represented by x and y, 

 we say that y is changing geometrically with x when the two 

 variables are following an equation of the form 



in which equation A = lO'', and B = lo^. By taking logarithms 

 of both sides of this equation, we may write 



log y = a -{- bx 

 and this equation shows that the two variables x and log y are 

 related in a linear fashion. This fact has led to the develop- 

 ment of a special form of graph paper on which one scale used 

 for X is arithmetic and the other scale used for 3; is logarithmic. 

 Any set of observations that fall along a straight line when 

 plotted on such paper will follow the equation of geometric in- 

 crease. 



The mode of derivation and the significance of the constants of 

 the equation will be given in the following example : 



In an article by Hartman (1927) is given a set of measurements 

 of the average area in square microns of twenty Plasmodia prcecox 

 in the asexual stage, taken at approximately hourly intervals 

 from 7.30 P.M. of one day until 4.00 p.m. of the following day. 

 These parasites change in size as is indicated in the following 

 table: 



