30O 
MATH'EMATICS: T. H. GRONWALL Proc. N. A. S. 
It is believed that the average probable error of the paralaxes of these 
nine clusters is much less than 15%. Suppose, however, that for one 
oi the clusters the true distance differs by 30% from the adopted value, 
lihe adopted absolute magnitudes for that cluster would be systematically 
in error by about 0.6 mag. The position of the maxima of the luminosity 
curves for the nine clusters together would be slightly altered, therefore, 
if the supposed error were uncompensated; but the general form of none 
of the curves would be materially changed. 
Figure 5 represents the distribution in color and luminosity of the 
individual stars, the circles representing data from Messier 11, the dots 
giving results for the most accurately measured clusters (Messier 3, 5, 
13, 15, and 68), and the crosses representing the stars for the clusters 
for which the magnitudes depend upon less extensive investigations. 
Together with the luminosity and color curves, this diagram illustrates 
the present state of our information concerning the giant stars in clusters. 
ON THE DISTORTION IN CONFORMAL MAPPING WHEN THE 
SECOND COEFFICIENT IN THE MAPPING FUNC- 
; TION HAS AN ASSIGNED VALUE 
\ By T. H. Gronwali. 
; Technicai. Staff, Office of the Chief of Ordnance, Washington, D. C. 
Communicated by K. H. Moore, April 27, 1920 
Note III On Conformal Mapping Under Aid of Grant No. 207 From the 
Bache Fund 
Let w = z -\- a^z^ + . . . + a„2:" + • • • be a power series in z convergent 
for 2^1 < |l and such that the circle \z\<\. is mapped conformally on a simple 
(that is, simply connected and nowhere overlapping) region in the w- 
plane. Koebe^ has shown that on the circumference \z\ = r, where 
0<f<l, the distortion [Jw/J^l and also 1^1 lie between positive bounds 
depending on r alone, and the writer^ has determined the exact values 
of these bounds. A far more difficult problem arises when some of the 
coefficients of the power series are given a priori. The simplest case Adhere 
a2 = ae'^^ (a^O) is given was investigated by the writer,^ the method 
employed failing, however, to furnish the upper bound of \z\ in the case 
0.5a <1. This defect has now been remedied, and denoting by r(a), 
f^r 0^a<l, the root between zero and unity of the equation 
- log ^ = 0, 
l + 2(a-l)r + f2 "1 
and by cos jS, for 0<f^f(a), the positive root of the equation 
2f 1 1 + ^ n 
log = 0, 
1 — 2rcos /3 + f ^ 1 — f 
wx have the following : 
