130 HOFMEISTER, ON 



the leaves.* Each cell of the second degree divides very 

 soon after its separation from the apical cell, by a septum 

 at right angles to the longitudinal axis of the stem, which 

 septum cuts the free outer wall, and also that lateral wall 

 of the cell which is turned towards the apical cell. 



* The first correct account of the cell-multiplication in the outermost apex 

 of the stem of Sphagnum was given by Niigeli ('Pflanzen. physiol-Unter- 

 suchungen,' i, Zurich, 1855, p. 76). I had previously ('Vergl. Unter- 

 suchungen,' p. 60) erroneously conceived the process to consist in the repeated 

 division of a two-surfaced, pointed, apical cell, by means of septa alternately 

 parallel to either of the two lateral surfaces. The origin of tiiis error was as 

 follows : When the arched apex of a very slender paraboloidal cellular body 

 consists of a single terminal cell (as is the case with the ends of the stein of 

 Sphagnum and Equisetum), a portion of the lateral edges of the apical cell will 

 usually be the only part clearly visible when the body is viewed from above. 

 The edges of the neighbouring cells of the second degree will not be seen. 

 These edges form arcs, the curvature of which is greater in proportion to the 

 size of the cells of the second degree, i. e. in proportion to the size of that por- 

 tion of the terminal cell which is cut off to form the cell of the second degree. 

 If the edges of the apical surface of the terminal cell of the bud extend so deep 

 down that at the spot where each two intersect the sides of the bud possess a 

 high degree of inclination, then, when the body is viewed from above, the 

 middle part only of each edge of the apical surface can be clearly seen. 



When the apical cell has the form of a three-sided inverted pyramid, with its 

 apical surface highly arched, and divides by septa arranged in a continuous 

 spiral order, and parallel to one of the lateral surfaces, then one of the edges of 

 the apical surface must, immediately after each division, be considerably shorter 

 than the two others. This fact is more clearly perceptible in proportion to the 

 size of that portion of the cell which goes to form the cell of the second degree. 

 In a system of similar spherical triangles, with a common centre, constructed by 

 drawing successively within each triangle arcs parallel to each one of its sides, 

 it will be found that one of the three arcs of each successive triangle is con- 

 siderably shorter than the other two, the difference being greater in proportion 

 to the curvature of the arcs, and to their distance from the respective parallel 

 sides of the next outer triangle (see the Diagram, PI. XVII, fig. 5*). When 

 the length of the arcs exceeds 90 ; when the length of the transverse diameter 

 of the outer surface of a cell of the second degree amounts to one half of the 

 diameter of the cell of the first degree from the division of which it originates ; 

 and lastly, when at the moment of division (by virtue of the inuate growing 

 power of the plant) the form of the apical surface of the cell of the first degree 

 is not that of an equilateral, but of an isosceles spherical triangle, then it may 

 happen that the points of intersection of the two larger arcs with the third 

 (very short) arc may fall quite outside the apical, vaulted surface of the organ, 

 when the latter surface is viewed under the microscope directly from above. 

 These remarks apply almost exactly to the apices of the stems of Sphagnum and 

 Equisetum, if observed immediately after the occurrence of division in the apical 

 cell. A figure of the upper surface of the terminal cell is then obtained, which 

 is strikingly similar to the apical aspect of the two-surfaced segment of a 

 spheroid (PI. XXII, fig. 4). Now, since in other instances (as in the apices of 

 the stems of liverworts, of Selaginella and of certain ferns, and in the organs 

 of fructification of mosses &c.) I had frequently ascertained that the multipli- 

 cation of the apical cell undoubtedly took place through division by means of 



