VARIATION— GENERAL RESULTS. 



25 



tainly presented the appearance of being- a teratological formation. In 

 the second place, it should be noted that while the end of the range is 

 theoretically at 1.14, the proportionate frequencies as given by the curve 

 for ordinates below that at 4.5 are excessively minute. Thus the abso- 

 lute frequency according to the theoretical curve for whorls of two 

 leaves is 0.00004.^ Or, in other words, the expectation of whorls with 



Leaf number 



Fig. 5.— Frequency histogram and fitted curve for variation in' leaf-number in Ceratophjilum. 

 All unmutilated whorls on plant 2, Series IV. 



two leaves given by the theoretical curve is that there will be 5 such 

 whorls in every 100, 000, 000! Finally, we must remember that the range 

 is subject to a very large probable error, both in respect to its absolute 



iThis is of course the area included in the range between 1.5 and 2.5, of which the 

 mid-ordinate is at 2, and not the ordinate at 2. To pass from ordinates to areas the 

 following formula, given by Pearson ( :04) for a curve of Type I was used. 



On strip of base h: 



h^ (mi + m; ) {x^ (mi + m; — 1) 



Area = /i X ^ 1 + 



[' 



^hcoT 



24 d/rf/ 



where h is the base element having y as its mid-ordinate at a distance x from the 

 mode, and d^ and rfj are the distances of y from the ends of the range. 



