68 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



for the law of growth here will be such a one that it will give reasonable 

 results for extrapolation at the upper end of the range. That is to say, 

 the same curve which gives a good graduation for the first 29 whorls, 

 ought, if it is to have any biological validity, to give a reasonable 

 result when extended to the 30th and further whorls. It is clear that 

 none of these parabolas is at all satisfactory in this respect. The gen- 

 eral failure of parabolas to give good extrapolation results, though they 

 may be entirely satisfactory for interpolation, has been recently empha- 

 sized by Miss Perrin ( :04) . The results obtained in the present case 

 form an excellent confirmation of her general position. On the whole 

 these results show very clearly that to put 



<^ (x) = Co + c^x + CiX^ + CsO;^ .... cX 



is inadequate in the present case. The form of the function which 

 expresses the true relation between size of whorl and its position is 

 something different from this. 



As has been mentioned above, inspection of the regression lines shows 

 that while the mean number of leaves per whorl increases the farther 

 out we go on the branch, yet at the same time the increment at each 

 successive whorl diminishes. This at once suggests another hypothesis 

 regarding the function. If we let dy denote a small change in mean 

 leaf -number and dx a small change in position, we may assume that 



^=1 .const. 



or in other words that the rate of increase in mean leaf -number varies 

 inversely as the position on the axis. This leads at once to a logarithmic 

 curve, which may be put in the following form. 



2/ = A + C log a?, 



where A and C are constants to be determined from the data, and, as 

 before, y and x measure respectively leaf-number per whorl and posi- 

 tion. In order to test the worth of our assumption it was decided to 

 try fitting a curve of this type. 



The same data as before were used. Before beginning the actual 

 calculation of the constants 'for the fitting, however, a graphical ''first 

 smooth" of the observations given in column 5 of table 34 was made on 

 a large scale and the smoothed ordinates read off. The values so 

 obtained furnished the raw material for the actual fitting. As before, 

 all the ordinates were given equal weight. Since we may obviously 

 take the origin of y (i. e. , of the ordinates) anywhere we please, it was 

 for practical reasons taken at 7 leaves. So that in the actual work each 



