ANNUAL GROWTH OF FRESH-WATER MUSSELS 
717 
The shells from the White River of Arkansas were collected by shell buyers 
during the summer of 1927. The collection consisted of 50 mixed left and right 
valves from males and the same number from females. 
The lower Rio Grande Valley shell collection was obtained from canals, settling 
basins, and “resacas” tributary to the Rio Grande during the winters of 1928-29 
and 1929-30. This Rio Grande collection consisted of 56 shells from males and 
26 from females. 
LENGTH IN RELATION TO AGE 
The maximum shell lengths included within each of the major interruption 
rings — that is, within the annual rings — having been obtained, the data were analyzed 
by the percentile method of Gal ton (1875). By inspection the 10 per cent groups 
seemed satisfactory for the various comparisons desired, consequently the first, 
fifth, and ninth deeds were computed for each set of measurements. The following 
formula has been used throughout this work: 
Deed = 
0 d-F ) i 
s 
+ v 
in which d equals coefficient of deed, that is, the number of cases considered times the 
numerical percentage determining the deed; F, the sum of the frequencies below the 
class in which the coefficient of the decil is located ; i, the class interval ; /, the frequency 
of the cases within the class in which the coefficient of the decil is located; and v, 
the value of the lower boundary of the class in which the coefficient of the decil is 
located. The probable errors for the first and ninth decils were computed by the 
formula : 
■d , , , 0.6745 iV. 09 N 
rrobable error = j' 
and for the fifth deed by the formula: 
t, , , , 0.6745 i^~N 
rrobable error = — 
These formulse for the probable error can, of course, be evolved from the formulae 
for standard deviation and mean probable error as currently used in statistical 
studies. (See Davenport, 1914.) 
Using the fifth decil values from the above data, curves were drawn showing 
trends of growth for the yellow sand shell from the Mississippi River in Iowa, the 
Wfiite River in Arkansas, and the lower Rio Grande in Texas. First and ninth 
decils, bounding as they do the 80 per cent of the population which conformed most 
closely to the normal growth of the group, were also plotted. It was felt that the 
10 per cent on either side of the first and ninth decils included most of the abnormal 
cases and errors in age determinations. At the same time it was realized that each 
of these two groups of extreme cases would quite likely contain true maximum and 
minimum deviations from the normal. As individuals of extremely rapid growth are 
likely to be of interest in connection with any genetic work which may be done with 
fresh-water mussels, in the figure showing all decils for each locality, the maximum 
and minimum cases for each year class have been added. These are indicated by the 
symbol for the locality in question, with a line drawn through it. 
