200 
MATHEMATICS: I. A. BARNETT 
Proc. N. a. S. 
cell are dependent upon the dimensions of the latter, its plane of division 
and the size and location of the nucleus. Thus, in very small, isodiametric 
cells, having a large centrally located nucleus, the cell plate quickly inter- 
sects the walls of the cell, without any extensive laterial growth — ^by means 
of additional peripheral fibers — of the spindle. In somewhat larger ele- 
ments there is sufficient room for the process of cytokinesis to reach the 
'*halo" stage before the cell plate intersects the sides of the cell. Only 
in elongated or much flattened elements, i.e., cells which are not isodi- 
ametric, is it possible for the phenomenon of cell plate formation to pass 
through the "spindle," "disk," "halo" and "frame" stages and finally 
form two entirely separate aggregations of kinoplasmic fibrillae, the 
kinoplasmasomes . 
It is evident, accordingly, that the remarkable type of cell plate forma- 
tion which was described in a former note, is not an isolated or unusual 
phenomenon, but is of frequent occurrence in the somatic tissue of the 
higher plants, gymnosperms and angiosperms. It promises to be signifi- 
cant in any general discussion concerning the dynamics of cytokinesis and 
karyokinesis. 
^ Phenomena of cell division in the cambium of arborescent gymnosperms and their 
cytological significance, these Proceedings, 5, 1919 (283-285). 
2 Bailey, I. W., The significance of the cambium in the study of certain physiological 
problems, /, Gen. Physiol. Ined. 
3 The average length of the cambial initials in 152 stems of gymnosperms was com- 
puted as approximately 3400 micra; in 275 stems of dicotyledons as approximately 
600 micra. 
^ The word fibers is used without reference to whether the fibrillae are true threads 
or lines of protoplasmic streaming in the living cell. 
^ Treub, M., "Quelques recherches sur le role du noyau dans la division des cellules 
vegetales," Verh. K. Akad. Wetensch. Amsterdam, 19, 1879 (1-35). 
^ Strasburger, E., Zellhildung und ZelUheilung, 3 aufl, Jena, 1880. 
7 SchiirhofT, P., "Das Verhalten des Kernes im Wundgewebe," Beih. Bot. Central., 
19, 1906 (359-382). 
FUNCTIONALS INVARIANT UNDER ONE-PARAMETER CON- 
TINUOUS GROUPS OF TRANSFORMATIONS IN THE 
SPACE OF CONTINUOUS FUNCTIONS 
By I. A. Barnktt 
Department of Mathematics, Harvard University 
Communicated by G. A. Bliss, February 20, 1920 
1. Some general notions. — It is intended in this abstract to give a few 
examples of one-parameter continuous groups of transformations in func- 
tion space which are analogues of well-known groups in the space of a 
finite number of dimensions, and to exhibit in each case a functional in- 
variant in terms of which every invariant of the group (with suitable 
restrictions) is expressible. Some of these groups have already been con- 
