206 
MATHEMATICS: R L. MOORE 
groups of eight lying just as they escaped from the oogonial sac and conspicu- 
ously oriented with respect to each other. The fact that groups of ten or of 
two are as regularly oriented, would refute the suggestion; but in order to 
prove that the phenomenon is the result of a stimulus acting after the eggs 
leave the oogonium, a group of them were transferred to a watch crystal and 
mixed with the point of a needle until their relative positions were entirely 
changed. But when they germinated the characteristic orientation with 
respect to each other was found to be without an exception. 
The only apparent explanation of the group orientation is that of a diffusion 
gradient of some substance emanating from a growing spore, or of some sub- 
stance being used up by it. A continuation of the investigation of this prob- 
lem will be an attempt to discover a substance which can so affect the dividing 
nucleus of the egg cell that its unequal distribution on the sides of the cell 
will orient the axis of the spindle. The effect of bubbling carbon dioxide and 
oxygen through cultures should be tried as being the most probable factors 
involved. 
The substance or condition originating in the activity of adjacent spores 
which has so powerful an effect in orienting the first cleavage plane and in 
determining which cell shall become the rhizoidal cell has no power to cause 
any chemotropism of the rhizoids after they are started. No rhizoid has 
been found to have its direction modified by the presence of other spores 
adjacent to it. In the absence of any light stimulus the rhizoids continue in 
the direction that they take originally from the spore. 
1 Blaaw, A. H., Rec. Travaux Bot. Neerlandais, 5, 1908, (209-372). 
2 Child, C. M., Individuality of Organisms, 1915. 
3 Day, E. C, Bull. Mus. Comp. Zool. 53, 1911, (303-343). 
4 Farmer, J. B., and Williams, J. L., Proc. Roy. Soc, 60, 1896, (188-195). 
• 5 Kniep, G. and Minder, F., Zs. Bot., 1, 1909, (619). 
6 MacDougal, D. T. and Spoehr, H. A., Science, (N. S.), 45, 1917, (616-618). 
7 Mast, S. O., /. Comp. Neur. Psych., 17, 1907, (99-179). 
8 Randolph, F. A., and Peirce, G. J., Bot. Gaz., <0, 1905, (321-350). 
^Rosenvinge, M. L. K., Revue Gen. Bot., 1, 1889, (125-135). 
10 Stahl, E., Ber. deuts. hot. Ges., 3, 1885, (334-340). 
u Winkler, H., Ihid., 18, 1900, (297-305). 
ON THE MOST GENERAL CLASS L OF FRECHET IN WHICH THE 
HEINE-BOREL-LEBESGUE THEOREM HOLDS TRUE 
By Robert L. Moore 
Department op Mai;hematics, University of Pennsylvania 
Communicated by E. FI. Moore, April 15, 1919 
§1 A class L of Frechet^ is a set of elements such that (1) if P is an element 
of L and Pi, P2, Pz . . . is a countable^ sequence of elements belonging to 
L then the statement that P is the limit of the sequence Pi, P2, P3, . . . 
