Academy. I concluded from this work on selection, that from the descendants of a single specimen produced by binary division, lines could be distinguished that were hereditarily diverse as regards spine number and diameter. The present study indicates that these heritable diversities may have been due to changes in the volume of the chromatin. These chromatin-cytoplasmic studies also have a bearing on the selection work carried on by various investigators, notably by Jennings with Difflugia and by Root with Centropyxis. 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 chromatin 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 CONFORMAL MAPPING

By T. H. GRONWALL

RANGE FIRING SECTION, ABERDEEN PROVING GROUND

Communicated by E. H. Moore, December 2, 1918

Note I on Conformal Mapping under aid of Grant No. 207 from the Bache Fund.

where p > 0 and 0 and 0' vary with p subject only to the conditions

e being positive but arbitrarily small, then

w(z) — w(x') → 0 as p→ 0

uniformly in respect to 0 and 0'.

It should be noted that no assumption is made regarding the convergence of the power series at z = 1. For the proof, we write w(z) = w1(z) + w2(z),

z'

0

N

1

and since | z − z | = pleti - e0'i | ≤ 2p, |z|< 1 and | z' | <1, it follows

that

Introducing z = 1-pei, z' = 1 - pe'i, a simple calculation shows that for p < sine and observing the limitations governing and ',

Since Zna, converges, the right hand member in (2) may be made less than an arbitrarily small 8 for p < sine by taking N sufficiently large, and having fixed N, the right hand member in (1) may be made less than ♪ by taking p sufficiently small. Thus, for p sufficiently small, | w(z) — w(z′) | < 2 ô independently of 0 and 0', which proves the theorem.

ρ

Let us now assume in particular that w = w(z) maps the circle | z | < 1 conformally on a simple (i.e. simply connected and nowhere overlapping) region D in the w-plane, and that all points of D are within a circle of radius R (this latter condition can always be brought about by a linear transformation on w and the extraction of a square root'); then

na

converges2 (and is less than R2). Finally, suppose that w(z') approaches a limit wo when z' approaches unity on the real axis; our theorem then shows that w(z) approaches the same point wo on the boundary of D when z tends toward unity along any curve interior to both the unit circle and an angle less than formed by two straight lines through z = 1 and symmetrical in respect to the real axis. This proposition is usually derived, somewhat less directly, from the distortion theorem.3

Koebe, 1. c., and Study, Konforme Abbildung einfach zusammenhängender Bereiche, Leipzig, 1913.

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