22 
MATHEMATICS: T. H. GRONWALL 
Academy. I concluded from this work on selection, that from the descend- 
ants of a single specimen produced by binary division, lines could be dis- 
tinguished that were hereditarily diverse as regards spine number and diame- 
ter. The present study indicates that these heritable diversities may have 
been due to changes in the volume of the chromatin. These chromatin-cyto- 
plasmic studies also have a bearing on the selection work carried on by various 
investigators, notably by Jennings with Difflugia and by Root with Centro- 
pyxis. Arcella, Difflugia, and Centropyxis all belong to the same group of 
Protozoa, the fresh-water Rhizopods; but the nuclei can not be seen in either 
Difflugia or Centropyxis, and hence at least some of the results obtained by 
Jennings and by Root may have been due to changes in nuclear number and 
consequently in chromatin mass, rather than an hereditary change in the chro- 
matin itself as seemed probable. An increase or decrease in nuclear number, 
however, does not account for simultaneous and independent diversities such 
as Jennings found in Difflugia with respect to shell diameter and length of 
spines, unless the assumption is made that certain nuclei exert an influence 
upon certain shell characters and other nuclei upon other shell characters. 
Evidence was obtained from my studies of Arcella polypora that hereditarily 
diverse strains with respect to nuclear number and shell diameter could be 
distinguished within a single line. More data regarding this and other related 
problems are very desirable. 
A THEOREM ON POWER SERIES, WITH AN APPLICATION TO 
CON FORMAL MAPPING 
By T. H. Gronwall 
Range Firing Section, Aberdeen Proving Ground 
Gommunicated by E. H. Moore, December 2, 1918 
Note I on Conformal Mapping under aid of Grant No. 207 from the 
Bache Fund. 
Theorem: Given a power series w(z) = convergent for | z |< 1 and 
such that ^n\a„\^ converges, and writing 
z = l-pe^\ z' = \-pe^"\ 
where p > 0 and 6 and B' vary with p subject only to the conditions 
TT IT TT . TT 
e being positive but arbitrarily small, then 
w{z) — w(z') —^Oasp—^0 
uniformly in respect to 6 and 6\ ^ 
