380 HARPER— ORGANIZATION, REPRODUCTION 



As in P. Boryanum, the general arrangement of the cells in the 

 most common form of sixteen-celled colony of P. asperum is one 

 in the center with five around this and two cells in the third or outer 

 series, the i -f- 5 + lo group, as Nageli and Braun called it. Nageli 

 ('49, p. 93) quite fully described these conditions for the group and 

 by comparing the areas of the concentric circles, assuming the radial 

 diameters of the cells are the same, showed geometrically that these 

 cell numbers are what he calls the " most natural," that is, the cells 

 so arranged come the nearest to filling the spaces in the concentric 

 circles as well as being the ones most commonly found. I have 

 described this arrangement ('16) as coming the nearest to that of 

 a least-surface configuration for such a group of rounded cells 

 lying in one plane. As I shall further discuss in another connec- 

 tion, Nageli is, however, probably mistaken, at least for P. Borya- 

 num, as Braun's figures ('55) show, in saying that for eight cells 

 the arrangement 2 -|- 6 is much less common than the " more nat- 

 ural " arrangement i -(- 7. 



I shall follow the same method of numbering the cells here as 

 in the case of P. Boryanuui (Fig. 25, and '16, Fig. ib), making the 

 central cell No. i and proceeding outward as shown in the diagram 

 (Fig. 7). The colony is bilaterally symmetrical as in P. Boryanum, 

 the axis bisecting cells i, 4, 7 and 12 and passing through the sur- 

 face of contact of cells 2 and 6. For describing the structure of 

 the colony more conveniently we may here, as in the case of P. 

 Boryanum, call the ends of this axis of the colony its m and n poles 

 respectively. 



The central cell is in contact with five cells. The cells of the 

 second series are each in contact with six cells and the cells of the 

 outer series are alternately in contact with three and four cells. 

 The cell walls meet in threes on the principle of least surfaces, ex- 

 cept where cells 2 and 6 are in contact as a pair right and left of 

 the axis of the colony. In P. Boryanum this grouping of inter- 

 sections is universal in the more regular sixteen-celled colonies and 

 the paired contact of cells 2 and 6 is one of the most obvious dift'er- 

 ences between the two species in their intercellular relations. 



If the tips of the lobes of a cell of series II. be connected in 

 serial order, g'^, g", d'-\ d'~, by straight lines, we have a trapezoid with 



