92 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



main they were plants which had wintered over without breaking up (cf . 

 p. 15, supra). When this happens and vigorous growth begins again 

 in the spring, we should expect to find the plants bearing branches* of 

 high average length and having an unusually large proportion of all 

 their whorls borne on primary branches. Reference to tables 25 and 26 

 shows exactly this to have been the case in Series VI. But on long 

 branches according to our law of growth there will be a great preponder- 

 ance of 10-leaved whorls. Hence we see the reason for the result 

 previously obtained, viz, ' 'The polygon for Series VI is not quite smooth 

 and regular, yet differs from all the others in having as the most frequent 

 class 10-leaved whorls. ' ' Series V differed from all the other series in its 

 gross frequency distributions in showing an unusually high proportionate 

 number of 7-leaved whorls. This arises from the fact that the two 

 large plants in this series (1 and 7) , when the spring growth began, 

 threw out an unusually large number of new lateral branches (second- 

 aries, tertiaries, and quaternaries). At the time the collections were 

 made these branches had not attained any considerable length. Hence 

 we have a preponderance of short branches in this series, and, as indi- 

 cated by the law of growth, we should in consequence expect a consid- 

 erable increase in the relative frequency of whorls with low leaf-number, 

 which is just what we find. 



It hardly requires discussion to make it evident that the facts embodied 

 in our law of growth explain how the characteristic values and differ- 

 ences in the variation constants for different regions of the plant, as set 

 forth in an earlier section, arise. It was found that the older the portion 

 of the plant the higher was the mean number of whorls. But the reason 

 for this is now clear, since the older portions will have grown longer than 

 the younger and produced more whorls with high leaf-numbers. It will 

 be seen later that the differences in variation in different portions of 

 the plant are to be accounted for similarly. 



We may turn now to another point which arises in connection with 

 positional differentiation. 



*So far as I am able to judge from the present material there is apparently nothing 

 in Ceratophyllum when it grows in this way as a perennial corresponding to the 

 "localized stages" discussed by Jackson ('99), Cushman (:02, :03, and :04), Shull ( :05), 

 etc. The new growth in the spring appears to go on in precisely the same manner as 

 it would have if there had been no interruption. This is, of course, what would be 

 expected in a plant like Ceratophyllum. My material at present is not sufficiently 

 great to enable me to make a positive statement that no change whatever in the 

 growth curve takes place at points where new growth of this kind begins. So far, 

 though, I have seen no certain indication of such a change. 



