70 



VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



The graduation is clearly a remarkably good one. A mean error of 

 0.0004 between observation and theory, when we are dealing with 30 

 ordinates is clearly as low as we could expect to get, considering the 

 probable errors to which the individual observations are subject. There 

 can be no doubt that we have found the mathematical expression of the 

 law according to which growth in the character under consideration 

 takes place. 



Tab-le 35.— Comparison of observed and calculated mean leaf-number for successive 

 whorls of primary branches. Series I, II, and III combined. Logarithmic curve. 



The curve and the observations are shown in plate i. 



This result I believe to be of very considerable interest and signifi- 

 cance from several points of view. In the first place, it gives us a pre- 

 cise and unique formulation of a fundamental law of growth and differ- 

 entiation in Ceratophyllum. We now know the nature of the change in 

 successively formed whorls on the growing branch. Before proceeding 

 to state this law in words it should be remembered that the differential 

 coefficient of our equation is 



dy _ 1 

 dx X — 



where the constant is, of course, 



C 



const. 



log,, 10 

 Or the law of change in successively formed whorls on primary branches 

 in Ceratophyllum may be stated in the following way: The mean num- 



