96 MEMOIRS OF THE NEW YORK BOTANICAL GARDEN 



the least possible disturbance in the swarming period will permit 

 the achievement of anything like a symmetrical arrangement. 

 I have already (2) indicated the method by which the configuration 

 of the colonies is determined and shall describe elsewhere the 

 stages in their development with the evidence that the two-lobed 

 form of the cells is a character which is inherited by the cells and 

 comes to expression in some degree at least quite independently 

 of their position in the colony. 



We are concerned now with the question as to the relation of 

 this more or less symmetrical organization of the colonics as we 

 find them to the physical principles illustrated in the formation 

 of least surface groups. It is obvious that the visible space rela- 

 tions of the cells to each other and to the organization of the whole 

 are simple and easily ascertainable. A drawing which represents 

 as far as possible a full expression of the tendencies which can be 

 discovered in the actual colonies is shown in figure lb. 



I have aimed in this diagram to eliminate all the irregularities 

 in position, size of cells, angles, etc., seen in the actual colony. 

 We can approximate such an ideal figure by taking the a\'erage of 

 the corresponding lines and angles right and left of the median axis 

 as they are found in a specimen selected for its regularity. The 

 dimensions of the lines and angles in this figure are obtained by 

 averaging those in the colony shown in figure la. 



The fluctuations in any particular dimension vary rather 

 symmetrically about a mode common to the whole group and to 

 a considerable degree balance each other. Figures of aberrant 

 individuals and statistical data as to the fluctuating variai)ility 

 in the grouj) will be published elsewhere. 



As was noted by Braun (1) the least surface grouping to which 

 the sixteen-celled colony tends to conform is that of nineteen 

 circles forming two series concentric about a central circle. If 

 these circles are flattened against each otluT we have the familiar 

 honeycomb configuration in which all boundaries meet in threes 

 and each included angle is I20°. 



The conspicuous feature of the sixteen-celled group is that while 

 showing a concentric arrangement of the cells it is also bilaterally 

 symmetrical, its axis bisecting cells i, 4, 7, and 12 and passing 

 through the surface of contact of cells 2 and 6. Except for the 

 slight reentering angle on the side toward cell 4 the central cvW is a 

 pentagon bounded by the bases of the five cells of series 2. 



