VARIATION AND CORRELATION
IN THE CRAYFISH

WITH SPECIAL REFERENCE TO THE INFLUENCE OF

DIFFERENTIATION AND HOMOLOGY

OF PARTS

BY

RAYMOND PEARL AND A. B. CLAWSON

WASHINGTON, D. C.

PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON

1907

CARNEGIE INSTITUTION OF WASHINGTON
PUBLICATION No. 64

? //- £ 0

PRINTED AT THE UNIVERSITY OF CHICAGO PRESS, CHICAGO

VARIATION AND CORRELATION IN THE CRAYFISH, WITH
SPECIAL REFERENCE TO THE INFLUENCE OF DIFFERENTIA-
TION AND HOMOLOGY OF PARTS.

BY RAYMOND PEARL AND A. B. CLAWSON.

INTRODUCTION.

The general purpose with which this study was undertaken was to
investigate by biometrical analysis the laws of variation in a number of
organs which, while serially homologous, were at the same time differ-
entiated among themselves in varying degrees. Such systems of organs
are of course well known. A very clear example is afforded by the
appendages of a crustacean, say a decapod. In such an organism there is
strict serial homology between the segments of the different appendages.
At the same time, a given segment of any single appendage is distinctly
differentiated from the homologous segments of the others. The primary
and immediate aim of the present study was to obtain as definite answers
as possible to the following two questions:

(i) What relation exists between the relative degree of variation
exhibited by any particular organ or character and its degree of differen-
tiation or specialization, as compared with other similar organs in a
homologous series?

(n) What relation exists between degree of correlation on the one
hand, and, on the other hand, the similarity of structure and position
indicated in the homology of parts in such a series of organs as is pre-
sented, for example, by the segments of the crustacean appendage?

Naturally the data collected to answer these questions threw light on
other problems of variation, which will be considered further on in the
paper. Both of the main problems we had before us have been incident-
ally considered in several cases by earlier workers, but, so far as we are
able to learn, no systematic investigation, aimed at their elucidation and
working by quantitative methods on a considerable number of characters
of an organism, has ever been made. It seemed desirable that such an
investigation should be undertaken.

1

2 VARIATION AND CORRELATION IN THE CRAYFISH.

In looking about for an organism suitable for such a study, the
common freshwater crayfish at once occurred to one as an especially
favorable form. In the crayfish there is no doubt about the serial
homology (attaching to the term the significance which it ordinarily
has in comparative anatomy) of the several joints of the successive ambu-
latory appendages. The most anterior leg (the cheliped) is widely dif-
ferentiated from the more posterior walking legs, among which as a class
there is little differentiation. Thus we are enabled easily to compare the
relative variability and correlation in slightly differentiated and greatly
differentiated homologous parts. Further, the firm exoskeleton makes it
possible to take measurements with a maximum of accuracy. Finally,
it is possible to obtain fairly large numbers of adult individuals with
relative ease.

The present paper forms one of a series in course of publication by one
of the authors, all of which have as their general aim the analysis of the
factors of variation and correlation, as a contribution toward the eventual
determination of the fundamental physiological (morphogenetic) laws
which underlie these phenomena. In different papers of the series the
attempt has been made to analyze by experiment and observation, sup-
plemented by biometrical methods, the influence of different factors on
the degree and nature of the variation exhibited by particular sets of
organs or characters. The number of factors which conceivably may
play a part in influencing variation is of course very great. From the very
complexity of the problem we are forced to the admittedly slow process
of making detailed and thorough studies of but few of these factors at
a time. The present paper deals with the influence on variation of
two such conceivable factors, namely, the degree of differentiation or
development of parts fundamentally similar in their structure, and the
relationship between such parts expressed in the term "homology."
In other papers of the series which either have appeared or will be pub-
lished shortly other sets of factors have been considered. The under-
lying ideas which have been in the mind of the writer of this paper
throughout the course of his work on variation are (a) that the phe-
nomena of continuous variation must themselves be thoroughly analyzed
before we can get at the laws which underlie them, and (b] that the
most certain way to make progress towards the desired goal is by follow-
ing the method of quantitative analysis. As the work has progressed,
the more firmly has the writer been convinced of the essential truth of
these ideas.

INTRODUCTION. 3

Having made clear the general problems with which this investigation
has to do it will not be necessary to state here the specific subsidiary
questions which we have attempted to answer. Instead these questions
will be stated in the body of the paper in direct connection with the data
bearing upon them.*

MATERIAL.

As material for this study a collection of crayfish belonging to the
Zoological Museum of the University of Michigan was used. We have
to thank Mr. C. C. Adams for kindly placing the material at our dis-
posal. The collection included about 450 adult individuals of both
sexes, all clearly belonging to the same species. The specimens were
collected from the River Rouge, near Birmingham, Michigan, on July
24, 1903, by Mr. J. B. Field, and were by him presented to the Univer-
sity Museum. They may be considered to form a homogeneous sample of
the crayfish population of that locality. The specimens belonged to the
species Carnbarus propinquus Girard. We have, then, a homogeneous
sample of material belonging to the same species, all the individuals of
which have probably been exposed to reasonably the same set of environ-
mental factors during their development. In the collection males were
about twice as numerous as females. We had at first intended to study
the variation in both sexes, but on account of the relatively small number
of the females, as well as for other reasons which need not be entered
into, it was decided to confine the attention to the males. All specimens
were discarded in which any one of the joints of the legs which were
chosen for measurement was either lost or undergoing regeneration.
There remained 283 normal males available for measurement. All these
males measured belonged to the so-called "Form I." It was apart from
the purpose of the present study to consider the relation of the dimorphism
of the males in the genus Carnbarus to variation, and hence only one form
was used in the investigation. This dimorphism has been discussed by
Hagen (1870), Faxon (1884), Harris (1901), and others.

* At this point I wish to acknowledge my great indebtedness to the Carnegie Insti-
tution of Washington for grants, during the tenure of which this work was done. It is
impossible to make any adequate statement of the value of this aid in my biometric
work. Under the best of conditions research in this field is very laborious and time-
consuming, and without adequate material facilities to lighten the burden of compu-
tation, and uninterrupted time for work, it is practically impossible to carry through
any extensive plan of biometrical work in any reasonable length of time. It should be
stated that I am alone responsible for the actual writing of this paper. — R. P.

4 VARIATION AND CORRELATION IN THE CRAYFISH.

CHAEACTERS STUDIED AND METHODS USED.

In choosing characters for measurement two somewhat conflicting
things had to be kept in mind. In the first place, to get adequate evi-
dence on the problems attacked it was necessary that several characters
should be taken in each of several members of a homologous series of
organs. In the second place, practical considerations with reference to
the time which could be given to the work demanded that the total number
of measurements to be taken on each individual should not be too great.
After careful consideration of the matter it was decided that the joints of
the legs offered on the whole the best chance of getting light on the
relation of variation to differentiation and hornology. The legs are more
satisfactory appendages to deal with than the maxillipeds, because of their
greater size. Further, the morphological specialization of parts connected
with the performance of special functions does not extend so far in the
case of the legs.

FIG. 1. — Outline of leg of crayfish bearing great chela.

It did not appear necessary for our purposes to take measurements of
the same characters on both sides of the body. Consequently all measure-
ments of bilateral characters were taken on the organs of the right side
of the animal only. It will be understood without further reference to
the matter that throughout the paper all statements regarding the joints
of the legs are based on data from the appendages on the right side of
the body. On each of the first three legs (i. e., the cheliped and the
first two walking legs) the length of each of the three distal joints was
measured. These distal joints were taken rather than more proximal
ones, because of the greater degree of differentiation which they present.
In addition to the measurements on the legs, the length of the cephalo-
thorax and the breadth of the head were recorded. The following
detailed statements in connection with fig. 1 will make plain the exact
character of the measurements taken.

For verbal convenience we shall throughout the paper use the follow-
ing conventions in referring to the different legs: The leg bearing the
great chela will be designated as "leg i;" the first walking leg as "leg n;"
and the second walking leg as "leg in."

INTRODUCTION. 5

Using the preceding terminology, the measurements made may be
listed as follows:

(1) Length of the meripodite of leg i, from the edge of its proximal dorsal process

for articulation with the ischiopodite to the distal edge of the joint on the
median line. (Fig. 1, a-&.)

(2) Length of the meripodite of leg n. Measurements from the points correspond-

ing to those given in (1).

(3) Length of the meripodite of leg in. Measurements from the points corre-

sponding to those given in (1) and (2).

(4) Length of the carpopodite of leg i, from the proximal dorsal edge to the end of

the distal dorsal process, where this joint articulates with the propodite.
(Fig. l,d-c.)

(5) Length of the carpopodite of leg n. Measurements from the points corre-

sponding to those given in (4).

(6) Length of the carpopodite of leg in. Measurements from the points corre-

sponding to those given in (4) and (5).

(7) Length of the propodite of leg i, from the proximal dorsal process where it

articulates with the carpopodite to the extreme distal end. (Fig. l,/-gr.)

(8) Length of the propodite of leg n. Measurements from the points correspond-

ing to those given in (7).

(9) Length of the propodite of leg in. Measurements from the points correspond-

ing to those given in (7) and (8).

(10) Length of the cephalothorax, from the tip of the rostrum to the posterior

margin on the dorsal median line.

(11) Breadth of the head, between symmetrical points on either side in the cervical

groove just in front of the two lateral spines.

The dimensions were taken by means of a pair of fine-pointed dividers.
They were then read off on a steel scale graduated to fifths of a millime-
ter. All readings were made with the aid of a hand lens. The records
were made to tenths of a millimeter, and may certainly be considered
accurate to fifths.

In reducing the material we have determined the correlation between
every possible pair of characters. Since we are dealing with eleven

11X10
characters, the number of possible pairs is - — , or 55. From this

u

material we are enabled to analyze in detail the phenomena of correla-
tion in the eleven characters enumerated.

In calculating the constants of the frequency distributions we have
throughout used Sheppard's corrections for the moments. The work
of computation was much facilitated by the use of a large Brunsviga
arithmometer.

6 VARIATION AND CORRELATION IN THE CRAYFISH.

GENERAL DISCUSSION.

The measurements upon which this work is based are given in the
form of correlation tables in the Appendix (tables 22 to 32). The first
thing which impresses one in studying these tables is the fact that we are
dealing here with very high correlations, and almost perfect linearity of
the regression. High correlation denotes, of course, a relatively great
degree of constancy in the proportionality of the correlated parts. Our
tables show that in the crayfish, so far as the range of the characters we
have considered is concerned, there is very little variation in the array of
one variable associated with a given type of the other. This means that
there is a very strong tendency for definite and particular conditions of
the different characters to be associated together. The fact that the
characters which were chosen for this study are generally so highly cor-
related makes them especially favorable for the discussion of our prob-
lems, since high values for the coefficients of correlation give us low
values for the probable errors, and hence we shall be able to estimate the
probable significance of small differences with considerable accuracy.

It seems desirable to collect together in one place the constants
measuring the degree of variation and correlation exhibited by the
characters studied. These constants are given in tables 1 and 3. These
form the fundamental reference tables; the data from them will be used
in succeeding portions of the paper, to answer special problems. Table
1 gives the following variation constants for the distribution of each of
the eleven characters: Mean, mode, standard deviation, coefficient of
variation, moment coefficients, /3n /32, kurtosis (/S, — 3), skewness, and
modal divergence. The unit for the "physical" constants is 1 mm., while
for the algebraical constants the unit is the unit of grouping, the value
of which in millimeters is given in the second column of the table.

From table 1 we note at once the following general facts regarding
variation in the characters under consideration:

(a) The relative variability in proportion to the size of the thing
varying is of roughly the same order of magnitude in practically all the
characters studied. With the single exception of the great chela (propo-
dite of leg i), the coefficients of variation all fall between 11 and 14.3
per cent. The great chela is very significantly more variable than any
of the other characters. It is of some interest to compare our results on
this point with some of the data which have been given by other workers
for variation in other Crustacea. Table 2 makes possible such com-
parison, and from the data there given one would conclude that in the
dimensions of the body the crayfish is proportionately a much less variable
form than is Eupagurus, and slightly more variable than Gelasimus.

GENERAL DISCUSSION.

TABLE 1. — Constants of variation in the crayfish.

CHARACTER.

UNIT

OF

GROUP-
ING.

MEAN.

MODE.

STANDARD
DEVIATION.

COEFFICIENT

OF
VARIATION.

Cephalothorax.

Head

Meripodite:

Leg i

Leg ii

Leg in

Carpopodite:

Leg i

Leg ii

Leg in

Propodite:

Leg i

Leg ii

Leg in

mwi.
1

0.5

0.5
04
0.4

0.4
0.2
0.3

1

0.3

0.4

26.032 ±0.115

10.554 ± .054

10.074 ± .053

7.847 ± .040

8.819 ± .042

7.3G8 ± .042

4.800 ± .026

6.139 ± .032

20.738 ± .150

7.060 ± .034

9.090 ± .043

24.743
10.318

9.6t36
7.492
8.486

7.037
4.562
5.873

19.189

6.855
8.983

2.865 ±0.081

1.347 ± .038

1.325 ± .038

1.006 ± .029

1.056 ± .030

1.051 ± .030

0.658 ± .019

.809 ± .023

3.743 ± .106

0.853 ± .024

1.072 ± .030

11.006 ±0.316
12.769 ± .368

13.151 ± .379
12.821 ± .369
11.972 ± .344

14.267 ± .413
13.705 ± .396
13.184 ± .380

18.047 ± .528
12.082 ± .317
11.789 ± .339

CHARACTER.

Cephalothorax.

Head

Meripodite:

Leg i

Leg ii

Leg in ....
Carpopodite:

Leg i

Leg ii

Leg in

Propodite:

Leg i

Leg ii

Leg in

8.2092
7.2594

7.0208
6.3258
6.9680

6.9056

10.8168

7.2794

14.0072
8.0844
7.1764

14.1666
8.0563

9.4291

8.9834

10.2582

9.9654
21.5662
16.7523

35.1225

10.9026
9.0192

220.7255

186.8628

153.5801
126.8318
159.39995

155.1885
385.5747
246.7453

661.1373
216.6607
163.7572

0.3628
.1697

.2569
.3188
.3110

.3016
.3675

.7276

.4489
.2250
.2201

3.2753
3.5459

3.1158
3.1696
3.2830

3.2543
3.2954
4.6565

3.3697
3.3150
3.1797

CHARACTER.

KURT09I3.

SKEWNESS.

MODAL DIVER-
GENCE.

TABLE NUMBER.I

Cephalothorax.

Head

Meripodite:

Leg i

Leg ii

Leg in ....
Carpopodite:

Leg i

Leg ii

L^ in

Propodite:

Leg i

Leg ii

Leg in

0.2753
.5459

.1158
.1696
.2830

.2543

.2954

1.6565

0.3697
.3150
.1797

+0.4500

+ .1748

+ .3077

+ .a530

+ .3158

+ .3144

+ .3619

+ ,3293

+ .4139

+ .2406

+ .0994

1.289 ± 0.141

0.236 ± .066

.408 ± .065

.355 ± .049

.333 ± .052

± .052

.238 ± .032

.267 ± .040

1.549 ± .184

0.205 ± .042

.107 ± .053

.22
.22

.22
.22
.22

.22

.22
.22

.22
.22

.22

i In this column are given the numbers of the correlation tables where will be found the dis-
tributions from which the constants of table 1 are deduced.

8

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 2. — Coefficients of variation in Crustacea.

ORGANISM.

CHARACTER.

COEFFICIENT
OF VARIATION.

AUTHORITY.

Eupagurus prideauxi:

Deep water, male

Shallow water, male.

Deep water, male

Shallow water, male.
Gelasimus pugilator:

Lefts, male

Do

Do..

Do.

Do.
Do.
Do.
Do.
Do.
Do.
Do.

Do.

Palaemonetes vulgaris

Both sexes

Do

Do

Do..

Right chela length

do do. . .

Carapace length . .
. .do. . . do. .

Frontal breadth

Median length

Lateral margin, great che-

lar side

Lateral margin, small che-

lar side

Meripodite of great chela.
Carpopodite of great chela
Propodite of great chela. .
Meripodite of small chela
Carpopodite of small chela
Propodite of small chela. .
Meripodite of second leg,

great chelar side

Meripodite of second leg,

small chelar side

Dorsal spine of rostrum . . .
Ventral spine of rostrum .
Dorsal spine of rostrum . .
Ventral spine of rostrum.

26.313 ±0.609
23.434 ± .502
19.446 ± .438
17.746 ± .372

7.471

7.598

8.880

9.000

9.657
10.472
10.255

9.920
11.139

8.898

9.660
9.160

9.83
15.03
IIO.OO
28.26

Schuster (1903)
Do.
Do.
Do.

Yerkes (1901)
Do.

Do.

Do.
Do.
Do.
Do.
Do.
Do.
Do.

Do.

Do.

Duncker (1900)
Do.
Do.
Do.

(6) The variation in all the characters shows a distinct skewness in
the positive direction. The value of the constant measuring the skew-
ness ranges from 0.10 to 0.45, being lowest in the case of the propo-
dite of leg in, and highest for the length of the cephalothorax. That
the distributions all deviate significantly from the symmeti'ical conditions
given by the normal or Gaussian law is evident if we compare the values
of the skewness and the modal divergence with the probable errors of
these constants for the normal curve. In a normal distribution where N,
the total number of individuals, is 283 we have

Probable error of skewness = ± 0.0491

The probable error of the "modal divergence" for each case, on the
assumption of normality, is given in table 1. From these values we
see that in only one case (the propodite of leg in) is the value of the
skewness or the modal divergence less than three times the probable
error. For this character the constant is in each case almost exactly
twice the probable error. Now, if these distributions were to be regarded

GENERAL DISCUSSION.

9

60

*5

35

f
I"

IS

10

t

/

\

/

N

\

/

\

/

•b

\

/

i1

1

\

\

s

•/-

S

X

f

/

V.

^

=i__

18.7 19.7 20.1 21.1 22.1 23.1 24.7 25.7 267 21.1 26.7 cS.7 30.7 31.7 3Z.1 33.7 3*.7 35.7 36.7 37.7 23.7

Length in millimeters
FIG. 2.— Frequency polygon and its fitted curve for variation in length of cephalothorax.

95

30

Th

\

/o

//.6 /?* /3.6 14.6 I5.S I6£ '7.6 (R6 19.6 20.6 Z/.6 ??.6 £3.6 &.6 25.6 26.6 ?76 ^a6 29.6 30.6 31.6 3Z.6 33.6 34.6 35.6

Length in millimeters
FIG. 3.— Frequency polygon and its fitted curve for variation in propodite of leg i.

4-0

L

\

7

10

S

\

4.5 49 5.3 5.7 6.1 6.5 6.9 7.3 7.7 8.1 6.5 8.9 9.3 9.7 /O.I 10.5 '0.9 11.3 11.7 12.1

Length in millimeters

FIG. 4. — Frequency polygon and its fitted curve for variation in carpopodite of leg I.

10

VARIATION AND CORRELATION IN THE CRAYFISH.

as following the normal law within the limit of errors due to random
sampling, neither of these constants should differ from zero by at most
more than twice the probable errors given. But the values of the con-
stants actually obtained, with a single exception, greatly exceed this
limit. Hence there can be no doubt that we are dealing here with sig-
nificantly skew variation. In order to make plain the character of these
distributions with which we are dealing, figs. 2, 3, and 4, have been pre-
pared. They show the frequency polygons and their fitted curves for the
variation in the following characters: Fig. 2, length of cephalothorax;
fig. 3, propodite of leg i; fig. 4, carpopodite of leg I.

The curve for each of these three polygons is of Pearson's Type i,
having the range limited in both distributions. The equations to the
curves are as follows:

Length of cephalothorax, y — 40.0539 (1 -\- fi .. ) /I — „ 0009)

\ 2.6248 /. x \14.a542

Propodite of leg r,
Carpopodite of leg I,

„ = 30.9939(1+5-1

, = 43.3056 (l+6-2462,

57907

\ 4. 6647

35.9791

/ x \22.9167

\ 30.6862/

It is evident from the diagrams that the curves give excellent gradua-
tions, quite as good as could be expected when dealing with less than 300
individuals in the observational series. There are some points regarding
the curves which need especial mention. In the first place, we note from
the equations that the theoretical range is large in each case. The actual
values are as follows:

Length of cephalothorax, theoretical range = 33.7542 mm.
Propodite of leg i, theoretical range = 42.5581 mm.

Carpopodite of leg i, theoretical range = 14.7730 mm.

These ranges evidently exceed considerably those observed, but this
excess is practically entirely due to the extreme "tailing out" of the
theoretical curve toward large values, i. e., on the + side of the mode.
The truth of this statement is shown by the values of the following table :

LOWER (-) Ei

tD OF RANGE.

UPPER (+) Ei

ro OF RANGE.

Observed.1

Theoretical.

Observed. i

Theoretical.

Length of cephalothorax

mm.
19.7

mm.

18.978

mm.
36.7

mm.
52.732

Propodite of leg i

12.6

12.610

33.6

55.168

Carpopodite of leg i

4.9

2 4. 539

11.3

2 19. 312

1 The values in these columns are the centers of the extreme classes of the observed ranges.

2 It should be remembered that the apparent discrepancy between these values and those of a, and
a3 as given in the third equation supra arises from the fact that the constants of the equation are given
in units of grouping, while here they are in millimeters.

GENERAL DISCUSSION. 11

No great stress is to be laid on the great overestimation by the theo-
retical curves of the ranges on the plus side, for two reasons. On the one
hand, the constant a2, which gives the range on the plus side of the mode,
is subject to a large probable error. On the other hand, the total frequency
beyond the observed upper limit of the range as given by the theoretical
curve is so small as to be entirely negligible for all practical purposes.
This will be evident from mere inspection of the ends of the curves, with-
out going into the figures to prove it.

Inspection of the curves leaves no doubt as to the substantial deviation
of the distributions from the symmetrical condition of the normal curve.
That the skewness is positive for all the characters studied is directly con-
nected with the fact that there is a very high degree of correlation between
these characters. Given a positive skewness for one character, say length
of cephalothorax, and we should expect that all other characters of the
same organism which are closely correlated in size with the cephalothorax
would also exhibit positive skewness. This is evident if we consider that
positive skewness in the character mentioned merely means that there are
more individuals with the cephalothorax larger than the modal condition
than there are individuals with it smaller than the mode. But if deviations
from the mode in cephalothorax length have associated with them corre-
sponding deviations in the same direction of the other size characters,
clearly we should expect a similar sort of skewness in the frequency dis-
tributions of these other characters. In the present case we actually do
find a very high degree of correlation or association existing between the
different pairs, and hence that all should show skewness in the same
direction is no more than is to be expected.

The explanation of the fact that the direction of the skewness is positive
is possibly in part that individuals in different molts were included in the
sample. This, however, can at most account for only a minor part of the
observed condition. It seems to us idle to speculate as to the fundamental
causes concerned in the production of skew variation in cases of this kind.
We can only hope to determine them by experimental investigations
conducted ad hoc.

(c) The kurtosis is in every case positive, or, in other words, /32>3.
In order to get an idea of the relative significance of this constant it is
necessary to compare the value of /32 — 3 with the probable error of ft2 for
the normal curve. Working from the well-known formula we have, when
^ =283,

Probable error of 0 = ± 0.1564

12 VARIATION AND CORRELATION IN THE CRAYFISH.

We see at once that in the case of the distributions for breadth of head
and for the length of the carpopodite of leg in, there is a certainly signifi-
cant deviation from the mesokurtic condition of the normal curve. The
distributions for the propodites of legs I and n are probably significantly
different from the condition of mesokurtosis. In the other cases the values
of /32 — 3 are less than twice the probable error and hence taken singly
would have to be considered as probably not significant. Due weight
must, however, be given to the fact that all the distributions show devia-
tions from mesokurtosis in the same direction.

((/) Putting all the data together, we may safely conclude that the con-
tinuous variation in the characters of the crayfish which we have studied
can not be adequately described by the normal or Gaussian curve. Both
in respect to the symmetry of the variation about the mean or modal con-
dition and in respect to the kurtosis the crayfish distributions deviate in
a uniform manner from the normal curve. They demand skew curves for
graduation.

We have in table 3 the coefficients of correlation, together with their
probable errors, for every possible pair of the eleven characters studied.
It is at once evident that we are dealing here with very high correla-
tions. The lowest coefficient in the table is 0.84, and from this the
values run up to as high as 0.973. These high correlations are in accord
with what has been found by other workers who have studied correlation
in other Crustacea. (Of. Yerkes, 1901, and Schuster, 1903, for example.)
The correlations between the different characters of the crayfish here
studied are of the same order of magnitude as has generally been found
for the correlation between bilaterally homologous organs or characters.*

In order to show graphically the closeness to linearity of the regres-
sions between these different characters of the crayfish we have prepared
the diagrams shown in figs. 5 and 6. Fig. 5 shows the regression of the
length of the propodite of leg I on the length of the cephalothorax, and
fig. 6 the regression of the length of the meripodite of leg in on the
length of the meripodite of leg I. These two cases were taken at random
merely as samples of what holds generally for all the other characters
studied.

* Cf ., for example, a table of the coefficients for such bilateral characters given as
table in in Davenport (1903).

GENERAL DISCUSSION.

TABLE 3. — Coefficients of correlation in the crayfish.

13

CHARACTER.

Cephalothorax

Head

Meripodite:

Leg I

Leg II

Leg III

Carpopodite:

Legl

Leg II

Leg III

Propodite:

Legl

Leg II

Leg III

CEPHALOTHORAX.

1

0.9358 ± 0.0050

.9627 ±
.9314 ±
.9582 ±

.9485 ±
.8948 ±
.8590 ±

.9349 ±
.9535 ±
.9525 ±

.0029
.0053
.0033

.0040
.0080
.0105

.0051
.0036
.0037

HEAD.

0.9358 ± 0.0050

1

.9282 ± .0056

.9184 ± .0063

.9233 ± .0059

.9156 ± .0065

.8753 ± .0094

.8395 ± .0118

.9106 ± .0068

.9209 ± .0061

.9304 ± .0054

MERIPODITE OF
LEG I.

0.9627 ± 0.0029
.9282 ± .0056

.9665 ± .0026

.9686 ± .0025

.9588 ± .0032

.8913 ± .0080

.8652 ± .0101

.9439 ± .0044

.9578 ± .00X3

.9579 ± .0033

MERIPODITE OF
LEG II.

0.9314 ± 0.0053

.9184 ± .0063

.9665 ± .0026

1

.9729 ± .0021

.9502 ± .0039

.9083 ± .0070

.8632 ± .0102

.9440 ± .0044

.9696 ± .0024

.9559 ± .0035

CHARACTER.

MERIPODITE OF
LEG III.

CARPOPODITE OF
LEG 1.

CARPOPODITE OF
LEG II.

CARPOPODITE OF
LEG III.

Cephalothorax

Head

Meripodite:

Leg I

Leg II

Leg III

Carpopodite:

Leg I

Leg II

Leg III

Propodite :

Leg I

Leg II

Leg III

0.9582 ±
.92X3 ±

.9686 ±

.9729 ±

1

.9601 ±
.9021 ±
.8742 ±

.9539 ±
.9670 ±
.9736 ±

o.ooa3

.0059

.0025
.0021

.0031
.0075
.0095

.0036
.0026
.0021

0.9485 ± 0.0040

.9156 ± .0065

.9588 ± .0032

.9502 ± .0039

.9601 ± .0031

1

.9069 ± .0071

.8685 ± .0099

.9677 ± .0025

.9531 ± .0037

.9463 ± .0042

0.8948 ± 0.0080

.8753 ± .0094

.8943 ± .0080

.9083 ± .0070

.9021 ± .0075

.9069 ± .0071

1

.8888 ± .0084

.8932 ± .0081

.9036 ± .0074

.9027 ± .0074

0.8590 ± 0.0105

.8395 ± .0118

.8652 ± .0101

.8632± .0102

.8742 ± .0095

.8685 ± .0099

.8888 ± .0084

1

.8613 ± .0104

.8673 ± .0099

.8757 ± .0093

CHARACTER.

PROPODITE OF
LEG I.

PROPODITE OF
LEG II.

PROPODITE OF
LEG III.

Cephalothorax ,

Head

Meripodite:

Legl

Leg II

Leg III

Carpopodite:

Leg I

LegH

Leg III

Propodite:

Legl

Leg II

Leg III.

0.9349 ± 0.0051

.9106 ± .0068

.9433 ± 0044

.9440 ± .0044

.9539 ± .00 J6

.9677 ± .0025

.8932 ± .0081

.8613 ± .0104

1

.9506 ±
.9402 ±

.0039
.0016

0.9535 ± 0.0036

.9209 ± .0061

.9578 ± .00X3

.9696 ± .0024

.9670 ± .0026

.9531 ± .0037

.9036 ± .0074

.8673 ± .0099

.9506 ± .0039

1

.9610 ± .0031

0.9525 ± 0.0037

.9304 ± .0054

.9579 ± .0033

.9559 ± .0035

.9736 ± .0021

.9463 ± .0042

.9027 ± .0074

.8757 ± .0093

.9402 ± .0046

.9610 ± .0031
1

14 VARIATION AND CORRELATION IN THE CRAYFISH.

19.7 ^/.7 ^3.7 Z5.7 27.7 ^9.7 31.7

Length of Cepha/othorax in mm.

33.7

35.7

37.7

FlG. 5.— Diagram showing the regression of the propodite of leg I on the cephalothorax. The circles
connected by a zigzag line give the means of the arrays of propodite length for given types of
cephalothorax length. The straight line is the calculated line of regression.

The characteristic equations connecting the characters are as follows,
where PI denotes probable mean length in millimeters of the propodite
of leg i for a given cephalothorax length C; and Mm denotes probable
mean length .11 millimeters of the meripodite of leg in for a given length,
MI, of the meripodite of leg I :

= 1.2214C -11.057
= .7720Jfi+ 1.042

(i)

VARIATION AND DIFFERENTIATION.

15

I •

10

K

7.15

e./5

9.15

10.15

II. IS IZ.I5

of MJ in mm.

I3.i5

IS.IS

FIG. 6.— Diagram showing the regression of the meripodite of leg in on the meripodite
of leg I. Significance of the lines as in fig. 5.

There can be no doubt as to the linearity of these regressions.
With these general data tables in hand we may proceed at once to the
discussion of special topics.

VARIATION AND DIFFERENTIATION.

The question with which we have to do in this section is that of the
relation of variation to differentiation and morphological specialization.
In respect to all of the joints investigated, legs I, n, and in are differen-
tiated from each other. In the case of leg I this differentiation is of
course obvious. We have in the cheliped a part which has developed to
a high degree in comparison with its serial homologues, the walking legs.
This specialization in respect to size and form is associated with the per-
formance of an altogether different set of functions from those of the
walking legs. An examination of the following table shows at once that
there is a sensible, if less marked, differentiation between corresponding
joints of legs n and in.

TABLE 4. — Comparison of the mean length of corresponding joints in legs II and III.

[Means from table 1.]

JOINTS.

LEG II.

LEG III.

DlFFEEENCE.

Meripodit6

7.847±0.040

8.819±0.042

0.972±0.058

Carpopodits . .

4 800± .026

6.139± .032

1.339± .041

Propodite

7.060± .034

9.090± .043

2.030± .055

16

VARIATION AND CORRELATION IN THE CRAYFISH.

Each joint of leg in is larger than the corresponding joint of leg n,
by an amount which is in every case many times its probable error. Not
only are legs n and in differentiated from one another in absolute size,
but also in their proportions, as the following ratios show.

100 X Carpopodite

Meripodite

100 X Carpopodite

Propodite

100 X Propodite
Meripodite

, leg n =61.2 per cent.; leg in = 69.6 per cent.
, leg n = 68.0 per cent.; leg in = 67.5 per cent,
leg ii = 90.0 per cent.; leg in = 103.1 per cent.

How do these differentiated appendages compare with respect to rela-
tive variability ? Does the degree of variability run parallel to the degree
of morphological specialization, or does the reverse relation hold? To
show the bearing of our results on these questions, table 5 has been pre-
pared. This gives the coefficient of variation for each joint of each leg.
On account of the differentiation of the legs in absolute size it is idle to
use the standard deviation as a measure of comparative variability.

TABLE 5. — Comparative variability of different appendages — coefficients of

variation.

JOINT.

LEG I.

LEG II.

LEG III.

Meripodite

13. 151 ±0.379

12.821±0.369

11.972±0.344

Carpopodite

14. 267 ± .413

13. 705 ± .396

13. 184 ± .380

Propodite

18. 047 ± .528

12. 082 ± .347

11. 789 ± .a39

Mean

15.155

12.869

12.315

From this table we note the following points:

(a) Leg I is the most variable of the three; leg n stands next, and
leg in is the least variable, but the differences between the last two are

o

practically insignificant.

(fr) The differences between corresponding joints of the different legs
in respect to relative variability are, on the whole, small as compared with
their probable errors. The only marked exception is the propodite in leg I.

(c) The considerable excess of the mean variability of the joints of
leg i over the means for legs n and in is largely due to the great
variability of the propodite of leg I (i. e., the great chela). This is by
far the most variable of any of the characters studied. The high varia-
bility of the great chela has been noted by other students of variation in
the Crustacea (cf. Yerkes, 1901, and Schuster, 1903).

VARIATION AND DIFFERENTIATION. 17

From these results we conclude that the appendage which is most
highly developed and specialized morphologically is also relatively the
most variable. This may be taken as a confirmation of Darwin's law that
highly developed parts tend to be more variable than corresponding parts
of ordinary size. It is in agreement with Yerkes' (1901) results on
Gelasimus.

So far we have been considering the variability in proportion to the
absolute size. Another equally important point to be considered is the
variation, not in the absolute size of the parts, but in the proportions of!
the body. Thus it is conceivable that the great chela might be relatively
the most variable part of the body in its absolute size, and yet that the
proportionate length of this organ in relation, say, to the length of the
cephalothorax would be one of the least variable dimensions. Having
determined the relative variability of the parts in respect to absolute size,
we must next study the variability in respect to proportions. This we
may of course do by expressing each dimension in the form of an index.
Thus, we may express the length of each joint of the legs as a percentage
of the cephalothorax length, instead of in terms of millimeters or other
absolute units. The variation of these indices will then measure the
extent to which the proportions, as distinct from the size of the parts
of the body are varying. In order to determine the variation shown
by these indices we may resort to the theorem given by Pearson (1897).
He has shown that if we let ?13 denote the mean value of an index,
a?,/;»3; 213 the standard deviation of this index; and vl and v3 the coeffi-
cients of variation of xl and x3 and r,3 the coefficient of correlation between
the same characters, then

(i)

where m1 and m3 are the means of the characters xl and x3, and

Sis = «'is l/^i2 + v32 • - 2ri3 ViV3) (n).

In the present case it seems best to use the length of the cephalothorax
as the basis of reference in determining the relative proportions of the
different joints of the legs. That is, in the index fraction xl/x3, x3 will
denote in every case the length of the cephalothorax, while for xl will be
put successively the different joints of the different legs. Proceeding in
this way, and putting the result in each case in the form of a percentage,
we shall have the length of each joint of the legs measured in hundredths
of the cephalothorax length.

18

VARIATION AND CORRELATION IN THE CRAYFISH.

Working in this way from formula (i) above, we have for the mean
indices, or the mean proportions of the joints of the legs in comparison
with cephalothorax length, the results set forth in table 6.

TABLE 6. — Mean proportions of the different joints of the legs.

[Unit = 1 per cent, of cephalothorax length.]

JOINT.

LEG I.

LEG II.

LEG III.

Meripodite

38.63

30 11

33.86

Carpopodite

28 22

18 44

23 57

Propodite

79.15

27.11

34.91

By this table of mean proportions the differentiation between the legs
is again clearly shown. In passing it may be remarked that the values
for the proportions of the different joints of the legs as given in this table
will perhaps be useful for diagnostic purposes in defining the species,
since they represent the mean of a considerable number of specimens.
In this connection, however, the point brought out in a later section of
the paper, dealing with "index correlations" (pp. 40-45), must be kept
in mind.

Turning to the question of the relative variability in the proportions
of the different legs, we have the results given in table 7. The values
in this table are the standard deviations of the indices whose means are
given in table 6.

TABLE 7. — Variability of the proportions of the different joints of the legs.

[Unit = 1 per cent, of cephalothorax length.]

JOINT.

STANDARD DEVIATION.

Leg I.

Leg II.

Leg III.

Meripodite

1.516±0.043
1.461± .041
6.875± .195

1.4.33±0.041
1.150± .033
0.997± .028

1.170±0.033
1.593± .045
1.255± .036

Carpopodite

Propodite

Mean

3.284

1.193

1.&39

From this table the following points are to be noted:

(a) Leg i, the most highly developed and differentiated of the three

under consideration, is the most variable in its proportionate as well as

in its absolute size.

VARIATION AND DIFFERENTIATION. 19

(6) When the proportionate size is considered, the order of compara-
tive variability of the legs is I, in, and n, whereas when we deal with
absolute size the order is I, 11, and in, as shown in table 5 above. The
difference between legs n and in in mean variability is relatively greater
for the proportionate than for the absolute dimensions.

(c) The great excess in variability indicated by the mean of the "leg i"
column of the table is, as before, clearly due in large measure to the extreme
variability of a single joint, the great chela. However we measure this
joint it is far and away the most variable of the lot.

(d) The extremely low variability of all the appeiidage-cephalothorax
indices (with the exception of that involving the propodite of leg i) is
noteworthy. It is another expression of the fact of extremely high
correlation in the parts of the crayfish body, which we have already
seen. Schuster (loc. cit.) found a very low variability of the chela index
in Eupagurus.

Putting all our results together, we have seen that when we compare
the several joints of three differentiated but serially homologous append-
ages of the crayfish in respect to relative variability, it is found that the
leg bearing the great chela is, in all the joints studied, the most variable.
This is true whether we deal with the variation of the parts in their
absolute sizes or with the variations in their proportionate dimensions
referred to the cephalothorax length as a base. But it is obvious that
the leg bearing the great chela is the one of those studied which is most
widely differentiated from the primitive type of the decapod limb, and
is also most highly specialized for the performance of a particular set of
functions. We hence conclude that, in the present case at least, a rela-
tively high degree of morphological differentiation and specialization
has associated with it a relatively high degree of variability in the parts
concerned. This, so far, is merely a statement of a fact regarding the
external morphology of the crayfish, and involves no theory as to the
nature or cause of such a relationship. An obvious suggestion as to the
cause of the greater variation shown by leg i is that since this is the
appendage which is on the whole most liable to injury, it will most often
be in process of regeneration. It might perhaps be maintained that our
sample contained a considerable number of individuals regenerating in
this leg, and that the greater variation really arose because of the inclu-
sion of different regeneration stages. It is possible that a part of the
observed result is to be explained in this way, but in our opinion the
influence of the possible inclusion of regenerating individuals must be

20 VARIATION AND CORRELATION IN THE CRAYFISH.

altogether insignificant in bringing about our observed results. There
are several reasons for this opinion. In the first place, as has been stated
above (p. 3), before the measuring was begun the material was very
carefully examined and in every case where there was the slightest evidence
that regeneration in any of the appendages measured was going on, the
specimen was put aside and not included in the measuring. This would
at once exclude all but those in which regeneration was very nearly com-
plete, and that there should have been any considerable portion of the
sample in which the right chela was in such an advanced stage of regen-
eration as to be indistinguishable from the normal under careful examina-
tion, is very unlikely. Furthermore the facts that (i) each of the three
joints studied is more variable in leg I than in either of the two other
legs, and (u) that the propodite of leg I is much more variable than
either the meripodite or carpopodite of that leg when taken together,
seem to us very difficult if not impossible of explanation on the assump-
tion that the observed degree of variation in leg I is in any considerable
part the result of the inclusion of non-homogeneous regenerating material.
The matter may be looked at in another way. It might be assumed,
with a fair degree of probability, that in any random sample whatever of
adult crayfish taken from their natural habitat there would be a definite,
perhaps even considerable, proportion of individuals which had regene-
rated at some previous time one or both of the great cheke. If, then, it
be further assumed that there is a tendency for a regenerated structure to
be formed with less quantitative precision — that is, in the aggregate with
greater variation — than when it is originally formed in the normal on-
togeny, we have at once an explanation for the greater observed variation
of leg I as compared with legs n and in. The chief difficulty with this
hypothesis is that we are in absolute ignorance as to the validity of the
second assumption. No systematic biometrical study of the relative vari-
ability of original and regenerated structures has ever been made, though
the field to be opened up by such work is a most promising one. It leads
at once to the general problem of the relative precision with which dif-
ferent morphogenetic processes operate. I have discussed this problem
in some detail for the normal ontogeny* and have been able to demon-
strate for one form, at least, that parts successively produced tend to
become less variable with each successive formation. If the same law
should hold true for regenerated structures as for successive formations

*Cf. Pearl, Pepper, and Hagle, 1907.

VARIATION AND DIFFERENTIATION. 21

in the normal ontogeny, then clearly the second assumption made at
the beginning of this paragraph is incorrect, and the suggested explana-
tion of our present results would fail. It is impossible to reach any con-
clusion regarding the point until biometrical studies on regeneration have
been made.

One further point in this connection is of interest. There is no
reasonable doubt that the differentiated, specialized condition of the leg
bearing the great chela is phylogenetically a relatively late acquisition.
In other words, it is not a primitive morphological condition. But we
find that this part which has been modified most recently phylogeneti-
cally is also the most variable of the three appendages studied. In so far
we have direct statistical confirmation in a single case of the dictum that
phylogenetically young structures tend to be more variable than those
older arid more primitive. In our opinion, however, no particular stress
is to be laid on this fact. The real cause of the greater variability of the
cheliped appears to us to lie most probably in ontogenetic (in particular,
growth) rather than phylogenetic factors.

So far we have discussed the variation in the appendage as a whole.
We may now consider very briefly the different segments of the leg
separately with reference to their comparative variability. The data are
given in convenient form in tables 5 and 7. It has already been noted
that the most variable single joint of all those on which we have data is
the propodite of leg I. Leaving this out of account as a specialized organ,
and comparing the other segments, it is seen from table 5 that in both
legs II and in the carpopodite is relatively the most variable. Also in
the case of leg I the carpopodite is more variable than the meripodite. A
relatively high degree of variability of the carpopodite, as compared with
meripodite and propodite, is shown in Yerkes' (1901) figures for Gela-
simus pugilator. Our results thus stand in agreement with his on this
point. In legs n and in the meripodite and propodite are substantially
equally variable. The differences in the coefficients are almost certainly
insignificant in comparison with their probable errors. These are the
results when the variation is measured from the absolute dimensions. If
we consider the variation in the proportions of the different segments as
given in table 7 the results are not quite so regular. Leaving out the
great chela as before, we see that in legs n and in the meripodite is more
variable than the carpopodite. The difference is probably not significant
in the case of leg I, but it certainly is in leg n. In leg in the carpopo-
dite is again the most variable joint. It might be thought that the

22

VARIATION AND CORRELATION IN THE CRAYFISH.

terminal segment of a series such as that presented by the leg of the
crayfish would occupy an extreme position with respect to relative varia-
bility, but such does not appear to be the case. The propodite is neither
the most variable nor the least variable joint uniformly.

We may next examine somewhat more in detail the facts regarding the
skewness in the different segments of the appendages. In table 8 the
data are arranged in convenient form for comparison.

TABLE 8. — Skewness in variation of the different joints.

SEGMENT OP LEG.

SKEWNE38 (+ IN ALL CA9E9).

Leg I.

Leg II.

Leg III.

Meripodite

0.3077
.3144
.4139

0.3530
.3619
.2406

0.3158
.3293
.0994

Carpopodite

Propodite

Mean

.3453

.3185

.2482

We see from this table that —

(a) Considering the means of the columns, leg I shows the greatest
skewness in its variation, and there is a steady decrease as we pass to legs
II and in. Great weight, however, is not to be laid on this conclusion
drawn from the means, for the reason that it is obvious that the excess of
the mean skewness for leg I over that for leg n is due entirely to the influ-
ence of a single joint, the propodite of leg I. It was shown above that
the great chela was the most variable single segment, and it is now seen
that it also has the greatest skewness in its distribution. In other words,
the most specialized structure of those studied shows a maximum degree
not only of variability but also of skewness.

(6) Leaving the great chela out of account, the segment which in each
leg gives the distribution with the greatest skewness is the carpopodite.
The difference between carpopodites and meripodites in skewness is not
large, but since in every case it is in the same direction, it is probably
safe to conclude that there is a slight tendency for the former to have
generally the more asymmetrical distributions. Here, as in («), we
find degree of skewness and degree of variability going parallel, it having
been shown above (p. 16) that the carpopodite is the most variable of
the joints after the great chela is put aside.

(c) The low degree of skewness in the distributions for the propodites
of legs ii and in is noteworthy. In fact, the propodite distribution for
leg in is, within the limits of error from random sampling, symmetrical.

CORRELATION AND LOCATION OF PARTS. 23

Summarizing the results of the section: It has been shown that in
the system of organs given by the joints of the crayfish leg, the most
highly differentiated and specialized organ of those studied, the great
chela, exhibits the greatest amount of variation and the highest degree
of skewness in its variation. Aside from the great chela the most vari-
able single segment on each leg is the carpopodite. This joint also shows
the greatest skewness in its distributions. In the cases where the differ-
entiation between segments is not very great in amount, there is very
little difference in the relative variabilities. So far as our variation
results go they suggest the following conclusions:

(i) That relative degree of morphological development or specializa-
tion and relative degree of variability run parallel.

(n) That degree of skewness and degree of variation tend to run
parallel.

It is to be understood that these are only the statements of results
obtained from the present material. They are not offered as generaliza-
tions, but rather as suggestions of questions on which any additional data
will be welcome.

CORRELATION AND LOCATION OF PARTS.

The discussion of the correlation results may conveniently be begun
with the consideration of the problem of the influence of the relative
position of two organs on the correlation between them. Is the correla-
tion between two contiguous organs or parts in general greater than the
correlation between two organs more or less widely separated from one
another? Or, put more generally, is there any definite relation between
degree of correlation and relative location of parts within the organism?
It is obvious that a definite answer to this question would be of impor-
tance in any attempt to formulate a general theory of the origin and laws
of organic correlation. In their elaborate study of the correlation between
the different bones of the human hand Lewenz and Whiteley (1902) found
in several instances clear evidence of such a "rule of neighbourhood," as
they term it. Thus they find (p. 360) that:

Generally there is a "rule of neighbourhood," i. e., any bone is more closely cor-
related with a second of the same series than with any other from which it is separated
by that second. Speaking roundly this is true for both lateral and longitudinal series;
but there are apparently significant deviations from this rule, the most notable of which
are, perhaps, those of the distal phalanges, which on all of the fingers of both hands tend
to be more highly correlated with the metacarpal bones or the proximal phalanges than
with the middle phalanges. The middle phalanges, however, obey the general rule.

24

VARIATION AND CORRELATION IN THE CRAYFISH.

In his study of correlation in Gelasimus pugilator Duncker (1903)
finds evidence of a law of contiguity. He says (p. 319):

Der wie bei bilateral homologen Merkmalen uberhaupt Btets positive Korrela-
tionskoeffizient der untersuchten Paare homologer Dimensionen ist um so grosser, je
naher die gemessenen Organe einander liegen.

With our present material it is obvious that we may consider the ques-
tion of contiguity and correlation from two standpoints. First, we may
take the structures in contiguous or non-contiguous metameres, i. e., in
the antero-posterior series; and again, we may test the matter on the con-
tiguous and non-contiguous segments within the single metamere. This
last gives what may be called a lateral series.

Taking first the location of parts in successive metameres, table 9
has been arranged to bring together the results in most convenient form
for direct comparison. Since we have data from three legs, it is evident
that there are three possible combinations of pairs, viz, I with n, I with
in, and ii with in. In the second, third, and fourth vertical columns of
the table the correlation coefficients for the joints of these three combina-
tions of legs are given. The particular pair of joints to which an indi-
vidual coefficient belongs is given in the first column of the table. The
first joint of each pair as they are entered in this column always belongs
to the first leg of each of the three pairs at the heads of the columns.
Thus, to take a single example to illustrate how the table is to be read,
by looking at the point where the sixth row and the second column of
figures meet, the entry 0.9601 is seen; this is the coefficient measuring
the correlation between the carpopodite of the cheliped and the meripodite
of the second walking leg. Finally, in the last column of the table a plus
sign ( -\- ) is entered whenever the correlation between a contiguous pair
of legs is greater than the correlation between a non-contiguous pair.
When the correlation is greater for a non-contiguous pair a minus sign
( — ) is entered.

TABLE 9. — Correlation and location of segments.

SEGMENTS.

LEGS I AND II.

LEGS I AND III.

LEGS II AND III.

EXCESS.

Meripodite with meripodite. . .
Carpopodite with carpopodite .
Propodite with propodite

0.9C65± 0.0026
.9069 ± .0071
.9506 ± .0039

0.9686 ±0.0025
.8685 ± .0099
.9402 ± .0046

0.9729 ±0.0021
.8888 ± .0084
.9601 ± .0031

- +
+ +
+ +

Meripodite with carpopodite .
Meripodite with propodite. . .
Carpopodite with meripodite .
Carpopodite with propodite . .
Propodite with meripodite .. .
Propodite with carpopodite. .

.8943 ± .0080
.9578 ± .003:5
.9502 ± .0039
.9531 ± .0037
.9440 ± .0044
.8932 ± .0081

.8652 ± .0101
.9579 ± .0033
.9601 ± .0031
.9463 ± .0042
.9539 ± .0036
.8613 ± .0104

.8632 ± .0102
.9559 ± .0035
.9021 ± .0075
.9027 ± .0074
.0670 ± .0026
.8673 ± .0099

+ —

+ -
- +
+ +

CORRELATION AND LOCATION OF PARTS.

25

From this table it is seen at once that the results with homologous
joints are quite different from those with non-homologous. Considering
only the pairs of homologous joints at the upper end of the table, there is
only one case out of the possible six in which the correlation is higher
between the non-contiguous pair of legs. In that case the difference
between the two coefficients is altogether insignificant in comparison with
its probable error (difference ==0.0021 ± 0.0036). The "rule of neigh-
bourhood" thus evidently holds for the correlation between homologous
joints of the different legs. The result may be stated in the following
way: The correlation betiveen the homologous segments of two legs is
higher when these two legs belong to contiguous metameres than when
they are separated by an intervening metamere.

There are twelve chances of comparison when the correlations of the
non-homologous segments are considered. A reference to the last column
of the table shows that in six of these cases the correlation is higher in
the non-contiguous pair of legs; while in five cases it is higher in the
contiguous pairs. In one case there is practically exact equality between
the coefficients, the difference being only 0.0001. The indication clearly
is that, so far as the non-homologous joint correlations are concerned, it is
about an even chance whether the excess will be in favor of the contiguous
or the non-contiguous pairs of legs. Before definitely accepting this con-
clusion, however, it will be well to determine whether the plus ( + ) differ-
ences are in the aggregate larger than the minus ( — ) , or vice versa.
Further, it is clear that it will be better to compare the proportions of
the differences to their probable errors rather than to the absolute values.
Accordingly, in table 10 is given for each plus ( + ) and minus ( — ) differ-
ence in the lower half of table 9, the value of the ratio of the difference to
its probable error (i. e., Diff. /P. E.

TABLE 10.— Correlation differences, to show the relative significance of the
plus ( + ) and minus ( — ) entries in the last column of table 9.

Diff. / P.E.Diff

Plus (+).

Minus ( — ).

2.26

0.14

1.21

.42

2.98

1.98

2.42

7.16

0.42

5.13

1.74

'1.86

'2.76

1 Mean.

26

VARIATION AND CORRELATION IN THE CRAYFISH.

From this table it is quite clear that neither the plus ( + ) nor the minus
( - - ) set of differences can be discarded as probably merely a result of
random sampling. Instead it must be concluded that there are real and
definite exceptions to the rule of contiguity when the correlations of non-
homologous joints are considered. The most striking exceptions to the
rule are the correlations which involve as one variable the carpopodite of
leg i. Three out of the four of these correlations give minus ( — ) excesses
and two are very large, the differences ratios being 7.16 and 5.13, the
largest in the table. On the whole we may conclude that in the correla-
tion of non-homologous segments of legs belonging to different metameres
there is no definite evidence in favor of the rule of contiguity. The
most striking exceptions to the rule are the correlations involving the
carpopodite of the cheliped.

We may next consider the problem of the influence of position on cor-
relation within the metamere. The question here is: Are contiguous
segments of the same leg more highly correlated than non-contiguous
segments? To answer this table 11 has been prepared on the same
general plan as table 9. As before, the excess is designated plus ( + )
when the contiguous joint correlations are the larger.

TABLE 11. — Correlation and position within the same metamere.

LEG.

MERIPODITE WITH
CAKPOPODITE.

MERIPODITE WITH
PROPODITE.

CARPOPODITE
WITH PROPODITE.

EXCESS.

I

0.9588 ± 0.0032

0.9439 ± 0.0044

0.9677 ±0.0025

+ 4-

11

.9083 ± .0070

.9696 ± .0024

.9036 ± 0074

Ill

.8742 ± .0095

.9736 ± .0021

.8757 ± .0093

It is evident that we have here the exact opposite of the rule of con-
tiguity, except in the case of leg I. There the two correlations involving
the carpopodite are high, just as they were found to be in the cross-
correlations considered above. In legs n and in the non-contiguous joint
correlations are greatly in excess of the contiguous. Too much stress
can not be laid on this result, however, because of the fact that we are
dealing with only three joints. Both of the contiguous joint correlations
necessarily involve the carpopodite as one of the variables. Now, a study
of table 3 will show that generally the carpopodite correlations of legs
n and ill are low, irrespective of the relative positions of the joints cor-
related. On the other hand the carpopodite correlations of leg I are
relatively high. There is apparently some other special factor influencing

CORRELATION AND HOMOLOGY. 27

carpopodite correlations which is strong enough to outweigh any poten-
tial influence of position. The question as to whether the rule of con-
tiguity holds within the metamere can not be definitely settled until a
larger number of segments than is here considered has been dealt with
from each leg.

The results set forth in this section of the paper may be summarized
as follows: The rule that structures occupying positions within the
organism contiguous to one another are more highly correlated together
than non-contiguous structures is found to hold (with a single insig-
nificant exception) for the correlations of the homologous segments of
different legs; it also holds for about half of the correlations between the
non-homologous joints of different legs; it fails signally in those cor-
relations involving a carpopodite as one of the variables, both in the case
of the non-homologous joint correlations of the different legs and in the
case of the correlations of the joints of the same leg. These exceptions are
probably due to the influence of some special factor concerned in the
carpopodite correlations.

CORRELATION AND HOMOLOGY.*

The problems to be considered in this section are as to the influence of
morphological homology on the degree of correlation between parts. Are
homologous structures in general more highly correlated than function-
ally similar but non-homologous structures? Our data furnish abundant
material from which light may be gained on this and the subsidiary
questions which are suggested by it. We ate
may first attack the general problem accord-
ing to the following plans: It is evident that a b ° B
the same segments of the different legs are „ „ „

serially homologous. The correlations between °

these Will furnish data for the "homologous" FlG" '-For explanation see text,
side of the comparison. What shall be compared with these to make a
fair test? It is clear that there are several possible ways to proceed. The
correlation between joint a of leg A (fig. 7) and its homologue, joint a of
leg B, may be compared with the correlations (i) between joint «' of leg A
and joint b' of leg B; (n) between joint a of leg A and joint c' of leg B.

* A preliminary paper dealing with the subject taken up in this section has been
published by Clawson (1905). An entire recalculation of the constants after the publi-
cation of this preliminary paper led to the discovery of several minor arithmetical errors.
These have now been corrected with the effect of making the whole system of correla-
tions somewhat smoother, but without essentially affecting the conclusions.

28

VARIATION AND CORRELATION IN THE CRAYFISH.

Similar combinations may be made for joint a of leg B and joint a"
of leg c. The whole system so formed will afford a comparison between
(a) the correlation of a joint in a first leg with its homologue in a second
leg, and (/3) the correlation of the first joint with the non-homologous
joints of the second leg. It is clear that in such a system of comparison we
avoid entirely any possible conflicting influence of the two factors homol-
ogy and contiguity of parts within the metamere. In order to detect
what the relative influence may be of homology and contiguity in an
antero-posterior direction (i. e., between metameres), we have merely to
separate the data in the appropriate way in the tables. In all of these
tables a plus ( + ) entry in the column headed "excess" means that the
constants for a pair of homologous joints is greater than that for the corre-
sponding non-homologous pair. The data are given in table 12.

TABLE 12.
(A) MERIPODITES OF LEGS I AND II, AND II AND III (CONTIGUOUS METAMERES).

03

o
w

J

HOMOLOGOUS
JOINTS.

COEFFICIENT.

EXCESS.

COEFFICIENT.

NON-HOMOLOGOUS JOINTS.

/

+ ( 8.60)

0.8943 ± 0.0080

Meripodite i with car-

hH

(

popodite ii (ab1 ).

M

Meripodite i )

}

+ ( 2.07)

.9578 ± .00a3

Meripodite i with pro-

T3
53

with meripo- [

0.9665 ± 0.0026 (

podite ii («c' ).

05

dite it (aa').1 )

)

+ ( 3.54)

.9502 ± .0039

Meripodite n with car-

1— 1

(

popodite i (a 'b).

\

+ ( 4.41)

.9440 ± .0044

Meripodite n with pro-

podite i (a'c).

+ (10.55)

.8632 ± .0102

Meripodite ii with car-

H- i
1— 1
M

T3
ft
03

Meripodite n
with meripo-
dite in (a'

«" \

.9729 ± .0021 <

+ ( 4.25)
+ ( 9.19)

.9559 ± .0035
.9021 ± .0075

popodite in (a 'b").
Meripodite ii with pro-
podite in (a'c").
Meripodite in with

1— 1
i__t

)•

carpopodite n (a"br).

-H 1.79)

.9670 ± .0026

Meripodite ii with pro-

podite i (a"c').

(B) MERIPODITES F LEGS I AND III (NON-CONTIGUOUS METAMERES).

+ (9.94)

0.8652 ± 0.0101

Meripodite i with car-

i— 5

popodite ii (ab").

H- 1
1— 1

Meripodite i )

+ (2.61)

.9579 ± .0033

Meripodite i with pro-

T3

with meripo-

0.9686 ± 0.0025 «

podite in (ac").

a

dite ni(aa"). )

+ (2.18)

.9601 ± .0031

Meripodite in with

i— t

carpopodite i (a"b).

+ (3.42)

.9539 ± .0036

Meripodite in with

propodite i (a"c).

'Tho letters in parentheses in this and succeeding tables refer to fig. 7.

CORRELATION AND HOMOLOGY.

29

The results shown in table 12 are very regular. In every case the
correlation is greater between the homologous joints than it is between
the non-homologous, though some of the differences are absolutely small.
The figures in the "excess" column show the relative significance of
these differences in comparison with their probable errors. These fig-
ures are the values of the ratio Difference/Probable error of difference.
Taking into consideration the values of this ratio in the different cases
and also the fact that in every case the difference is in the same direction,
it is clearly justifiable to conclude that there is a uniform and definite
tendency for the meripodite of any leg to be more highly correlated with
the homologous segment (i. e., the meripodite of a second leg} than with
any other segment of that second leg. It is, of course, to be understood
that this statement is made only for those legs and joints studied in this
work.

We may turn next to the carpopodite correlations (table 13), treating
the data in the same way.

TABLE 13.

(A) CAKPOPODITES OF LEG I AND II, AND II AND III (CONTIGUOUS METAMERES).

i

HOMOLOGOUS
JOINTS.

COEFFICIENT.

EXCESS.

COEFFICIENT.

NON-HOMOLOGOUS JOINTS.

-(5.35)

0.9502 ± 0.0039

Carpopodite i with

1—5

hH

c

Carpopodite i '
with carpo-
podite n

i ht^ " \

0.9069 ± 0.0071 .

- (5.78)
+ (1-18)

.9531 ± .0037
.8943 ± .0080

meripodite ii (ba').
Carpopodite i with
propodite n (be').
Carpopodite ii with

(00 ).

meripodite i (b'a).

+ (1-28)

.8932 ± .0081

Carpopodite n with

propodite i (b'c).

/

-(1.19)

.9021 ± .0075

Carpopodite ii with

1— 1

1— 1
1— 1

0)

Carpopodite
ii with car-
popodite in

/ t\ ' /A " \

.8888 ± .0084 1

+ (1.94)

.9027 ± .0074
.8632 ± .0102

meripodite in (b'a").
Carpopodite ii with
propodite in (b'c").
Carpopodite in with

1— 1

(0 0 ).

(

meripodite n (b"a').

\

+ (1.67)

.8673 ± .0099

Carpopodite in with

propodite ii (b"c').

(B) CARPOPODITES OF LEGS I AND 111 (NON-CONTIGUOUS METAMERES).

1

- (8.89)

0.9601 ± 0.0031

Carpopodite i with

i— i

Carpopodite i
with carpo-
podite in

0.8685 ± 0.0099 1

-(7.27)
+ (0.23)

.9463 ± .0042
.8652 ± .0101

meripodite in (ba").
Carpopodite i with
propodite in (be").
Carpopodite in with

(bb ).

I

meripodite i (b"a).

\

+ (0.50)

.8613 ± .0104

Carpopodite in with

propodite i (b"c).

30 VARIATION AND CORRELATION IN THE CRAYFISH.

Table 13 shows perfect regularity in the trend of the "excess," but
it is of a quite different kind from that observed in table 12. Here
there is no uniform rule that the homologous joints show the higher
correlations. Instead there appears to be a special factor governing the
carpopodite correlations which leads to a rule which may be formulated
in the following way: (i) The carpopodite of any given leg is correlated
to a relatively high degree with the meripodite and propodite of each of
the legs which lie posterior to it, and further, the correlations of carpopo-
dite with meripodite and propodite are practically equal; (n) the car-
popodite of any given leg is correlated to a relatively low degree with
the meripodite and propodite of each of the legs anterior to it, and again
these correlations are substantially equal. Consequently we find that
the "homologous joint" carpopodite correlations are higher than the non-
homologous joint correlations when the latter are directed anteriorly (by
II ), and lower when they are directed posteriorly (by i). Consequently,
as a result of special factors influencing carpopodite correlations, the effect
of homology is in part overshadowed. It is clear from the figures in
parentheses in the excess column that the minus (--) differences are
relatively, as well as absolutely, greater than the plus ( -f- ) .

The data for the propodite correlations are given in table 14.

The results here are rather more irregular than one could wish. There
is clearly no uniform tendency for the homologous joint correlations to
be higher than the non-homologous. In both the contiguous metamere
and non-contiguous metamere groups there is a mixture of plus ( + ) and
minus ( — ) entries in the excess column. Consequently, in order to get a
notion of the general trend of the results, we are thrown back on an estimate
of the relative magnitude of the plus and minus differences. From the
figures in the excess column it is evident that on the whole the plus
differences run distinctly larger than the minus. The average of all the
plus difference ratios is 5.32, while for the minus ratios it is 1.62. From
this it appears reasonable to conclude that on the whole there is a distinct,
though very slight, tendency for the correlations of homologous propo-
dite segments to be higher than those of non-homologous joint pairs
involving a propodite as one variable.

Putting all the results set forth in tables 12 to 14 together, we may
conclude that in general, so far as indicated by our present material,
there is a slight but distinct tendency for homologous pairs of joints to
be more closely correlated than non-homologous pairs. In the case of

CORRELATION AND HOMOLOGY.

31

pairs involving a carpopodite as one variable this tendency is outweighed
by special positional influences affecting the variations and correlations of
this joint. In general the influence of homology on correlation is a
comparatively weak one — not nearly so strong as would probably have
been predicted.

TABLE 14.
(A) PROPODITES OF LEGS I AND II, AND II AND III (CONTIGUOUS METAMERES).

in
C
W

HOMOLOGOUS
JOINTS.

COEFFICIENT.

EXCESS.

COEFFICIENT.

NON-HOMOLOGOUS JOINTS.

H

+ ( 1-14)

0.9440 ± 0.0044

Propodite i with meri-

M

podite ii (ca').

HH

Propodite i)

+ ( 6.45)

.8932 ± .0081

Propodite i with car-

T3

Cj

with propo- (•

0.9506 ± 0.0039 .

popodite ii (cbf).

a

dite ii (cc'). )

-( 1-41)

.9578 ± .0033

Propodite n with meri-

H- :

podite i (c'a).

-( 0.47)

.9531 ± .0037

Piopodite ii with car-

popodite i (c'6).

ii^j

-( 1.50)

.9670 ± .0026

Propodite n withmeri-

t-H

podite in (c'a").

HH
rQ

Propodite n )

+ ( 9.01)

.8673 ± .0099

Propodite ii with car-

C

with propo- C

.9610 ± .0031 «

popodite in (c'6").

(0

^^

dite in (c'c"). )

+ ( 1.11)

.9559 ± .0035

Propodite in with meri-

HH

podite ii (c"a').

+ ( 7.29)

.9027 ± .0074

Propodite in with car-

popodite n (c"b').

(B) PROPODITES OF LEGS I AND III (NON-CONTIGUOUS METAMERES).

-( 2.36)

0.9539 ± 0.0086

Propodite i with meri-

I-H

podite in (ca").

HH
I— 1

Propodite i )

+ ( 6.92)

.8613 ± .0104

Propodite i with car-

T3

with propo- (

0.9402 ± 0.0046

popodite in (c6").

a

dite in (cc"). )

-( 3.16)

.9579 ± .0033

Propodite in with meri-

h- 1

podite i (c"a).

-( 0.98)

.9463 ± .0042

Propodite in with car-

popodite i (c"b).

The next problem to be considered with reference to the influence of
homology on correlation is as to the relative effect of homology and of
contiguity within the metamere on the degree of correlation. Is a given
segment in general more or less highly correlated with the homologous
segment in another leg than it is with the other segments (contiguous
and non-contiguous) of the leg to which it belongs itself? The data
on this question are presented in tables 15 to 17, which are arranged on
the same plan as tables 12 to 14.

32

VARIATION AND CORRELATION IN THE CRAYFISH.

From table 15 it would appear that, so far as meripodite correlations
are concerned, there is a distinct tendency for homologous joints of two
different legs to be more highly correlated together than are the different
joints of the same leg, whether contiguous or not. There are only three
exceptions to the rule, and in these cases the differences are insignificant
in comparison with their probable errors (cf. excess column).

TABLE 15.
(A) MERIPODITES OF LEGS I AND II, AND II AND III (CONTIGUOUS METAMERES).

OB

t3

i
3

HOMOLOGOUS
JOINTS.

COEFFICIENT.

EXCESS.

COEFFICIENT.

NON-HOMOLOGOUS
JOINTS OF SAME LEG.

+ ( 1.88)

0.9588 ± 0.0032

Meripodite i with car-

i— i

popodite i (ab).

i— 1

Meripodite i)

+ ( 7.76)

.9083 ± .0070

Meripodite n with car-

T3
C

with meripo- C

0.9665 ± 0.0026 ,

popodite ii (a '6').

a

dite ii (aa1 ). }

+ ( 4.43)

.9439 ± .0044

Meripodite i with

t— i

propodite i (ac).

-( 0.89)

.9696 ± .0024

Meripodite ii (with

propodite n (a'c').

+ ( 8.85)

.9083 ± .0070

Meripodite n with car-

hH
1— I

popodite in (a'b1).

H- 1

Meripodite n )

+ (10.18)

.8742 ± .0095

Meripodite in with car-

t3

c

with meripo v

.9729 ± .0021

popodite in (a"b").

a

dite in (aa"). )

+ ( 1.06)

.9696 ± .0024

Meripodite ii with

i— i

h- 1

propodite n (a'c').

-( 0.23)

.9736 ± .0021

Meripodite HE with

propodite HI (a" c").

(B) MERIPODITES OF LEGS I AND III (NON-CONTIGUOUS METAMERES).

+ ( 2.39)

0.9588 ± 0.0032

Meripodite i with car-

i—5

popodite i (ab).

t— i

H- 1

Meripodite i )

+ ( 9.63)

.8742 ± .0095

Meripodite HI with car-

T3

with meripo- [•

0.9686 ± 0.0025 .

popodite HI (a"b").

C

a

dite in (aa"). )

+ ( 4.84)

.9439 ± .0044

Meripodite i with pro-

h- 1

podite i (ac).

-( 1.52)

.9736 ± .0021

Meripodite HI with

propodite HI (a"c").

Table 16 demonstrates that in the case of the carpopodites, on the
whole, the contiguous joints of the same leg are more highly correlated
than homologous joints of different legs. In only three cases out of twelve
is the excess in favor of the homologous joints, and in those cases the
differences are relatively small. The carpopodite correlations thus show
exactly the reversed relation to what the meripodite correlations do,
though the differences, as shown by the ratios in the excess column,
are relatively smaller than in the former case.

CORRELATION AND HOMOLOGY.

33

TABLE 16.
(A) CARPOPODITES OF LEGS I AND II, AND II AND III (CONTIGUOUS METAMERES).

oi

i

rf

HOMOLOGOUS
JOINTS.

COEFFICIENT.

EXCESS.

COEFFICIENT.

NON-HOMOLOGOUS
JOINTS OF SAME LEG.

- (6.65)

0.9588 ± 0.0032

Carpopodite i with

h— I

1— I

T3

«

Carpopodite i
with carpo-
podite ii
(55')

0.9069 ± 0.0071

-(6.77)
-(0.14)

.9677 ± .0025
.9083 ± .0070

meripodite i (ba).
Carpopodite i with
propodite i (be).
Carpopodite n with

HH

yc'i/ j .

meripodite ii (b'a').

+ ( -32)

.9036 ± .0074

Carpopodite n with

propodite n (b'c').

-(1.79)

.9083 ± .0070

Carpopodite n with

.

meripodite n (5' a').

HH
H-l

Carpopodite

-(1.32)

.9036 ± .0074

Carpopodite n with

HH

T^

ii with car-

.8888 ± .0084

propodite ii (b'c').

u

a

popodite in

+ (1.15)

.8742 ± .0095

Carpopodite in with

t/S

(b'b').

meripodite in (5"a").

HH
M

+ (1.05)

.8757 ± .0093

Carpopodite in with

propodite in (b"c").

(B) CARPOPODITES OF LEGS I AND III (NON-CONTIGUOUS METAMERES).

-(8.68)

0.9588 ± 0.0032

Carpopodite i with

hM

I-H
1— I

T3

a

aS

Carpopodite I
with carpo-
podite in
thl>"\

0.8685 ± 0.0099 ,

-(9.73)
- (0.42)

.9677 ± .0025
.8742 ± .0095

meripodite i (ba).
Carpopodite i with
propodite i (be).
Carpopodite in with

1— I

\uu >•

meripodite in (b"a").

-( -53)

.8757 ± .0093

Carpopodite in with

propodite in (b"c").

We have, finally, the data for the propodites in table 17, and here we
find that in the case of the propodite correlations there is nothing approach-
ing a uniform rule of higher correlation in the homologous joint pairs
over the pairs from the same leg. In seven cases out of the twelve
the pairs of joints from the same leg have the higher coefficient, and in
five the opposite relation holds. No stress can be laid on such a small
majority, however. Turning to the amounts of the individual differences,
we find for the plus difference ratios a mean value of 5.76 and for the
minus ratios a mean of 3.70. The plus differences are clearly the larger
on the average.

34

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 17.
(A) PROPODITES OF LEGS I AND II, AND II AND III (CONTIGUOUS METAMERES) .

00

&

a
3

HOMOLOGOUS
JOINTS.

COEFFICIENT.

EXCESS.

COEFFICIENT.

NON-HOMOLOGOUS JOINTS.

-(3.72)

0.9677 ± 0.0025

Propodite i with car-

^•4

popodite i (c6).

h- 1

Propodite i )

+ (5.60)

.9036 ± .0074

Propodite n with car-

rd

g

with propo-

0.9506 ± 0.0039

popodite ii (c'b1 ).

3

dite ii (cc1). )

+ (1.14)

.9439 ± .0044

Propodite i with

i— i

meripodite i (ca).

-(4.13)

.9696 ± .0024

Propodite n with

meripodite ii (c'a1).

+ (7.18)

.9036 ± .0074

Propodite n with car-

h-5

popodite ii (c'b' ).

HH

h- 1

Propodite n )

+ (8.70)

.8757 ± .0093

Propodite in with car-

T)

with propo- [

.9610 ± .0031

popodite in (c"b").

d

a

ditem(c'c"). )

-(2.21)

.9696 ± .0024

Propodite n with

H- 1

meripodite ii (c'a').

>"i

-(3.41)

.9736 ± .0021

Propodite in with

meripodite in (c"a").

(B) PROPODITES OF LEGS I AND III (NON-CONTIGUOUS METAMERES).

/

-(5.29)

0.9677 ± 0.0025

Propodite i with car-

HH

(

popodite i (cfo).

hH
h- 1

Propodite i )

]

+ (6.20)

.8757 ± .0093

Propodite in with car-

13

with propo- C

0.9402 ± 0.0046 (

popodite in (c"6").

a

cd

dite in (cc"). )

)

-(0.58)

.9439 ± .0044

Propodite i with

(— 1

(

meripodite i (ca).

\

-(6.55)

.9736 ± .0021

Propodite in with

meripodite in (c"a").

Putting our results together we see that (a) in the case of meripo-
dites the influence of homology outweighs that of contiguity of parts;
(6) in the case of carpopodites the positional (contiguity) influence is
the greater; and finally (c) in the case of propodites there is a fairly
even balance between the two factors. Clearly it is impossible to lay
down any rule that in general the morphological relationship implied
in homology has a stronger influence in determining degree of correla-
tion between parts than does the relative position of the parts in the
organism (contiguity or separation). It would appear rather that relative
position of parts and homology are about equally effective in influencing
correlation. The important point, however, is that the influence of both
these factors is very slight. Our results indicate that both position and
homology are factors having a real influence on degree of correlation, but
the amount of these influences might, on general grounds, very easily
be — in fact probably has been — overrated.

CORRELATION AND HOMOLOGY. 35

While the figures set forth in the tables of this section show that in
general there is some tendency for homologous joint pairs to be more
highly correlated together than are non-homologous, yet they equally
show that this influence may be quite outweighed by special factors
influencing particular joints. The carpopodite correlations are clear
illustrations of this point. Now, there can be little doubt that these other
factors which come in to influence correlations are in general factors con-
nected with the functional relations of the parts ; that is, in a broad sense,
physiological factors. Perhaps the most obvious of such physiological
factors is growth. In the next section it will be shown that on the
average more than 50 per cent, of the observed gross correlations between
the joints of the crayfish appendages is due to a true growth correlation
factor. In comparison with such an effect as this it is obvious that the
influence of homology on correlation is practically a negligible one. From
these facts we are compelled to conclude that in comparison with physio-
logical factors the influence of morphological relationship, as implied in
homology, on correlation is relatively insignificant. This conclusion is
in entire agreement with certain results which have been obtained by
Davenport (1903, p. 130). Studying the correlation between the antero-
posterior diameter and the dorso-ventral diameter of the lower valve of
Pecten opercularis, he finds a very high degree of correlation between
them. These axes are morphologically independent, but by the position
which the animal takes they are brought into the same relation to the
bottom. The coefficient of correlation between these two axes is in each
of three samples > 0.969. Regarding this result Davenport says:

Here the correlation coefficients of non-bilateral dimensions are extremely high, as
high as in many of the highest human coefficients between bilateral dimensions. As
a result of the newly assumed position of the scallop, two formerly largely independent
axes have come to vary simultaneously just because they have similar relations to the
bottom. Pecten has gained a new kind of symmetry; namely, a radial symmetry.
The fact points very forcibly to another conclusion; namely, that physiological factors
are much more important in determining correlations than morphological relationship
when the two come into conflict.

Though the influence of homology on correlation is a relatively slight
one, that it is nevertheless a real one is evidenced by the fact that work
on other organisms than the crayfish has shown its existence. In his
memoir on the variation and correlation of the human skeleton Warren
(1897) includes a section on "The Correlation of Homologous Parts" (pp.
179-182) in which he discusses the relative degree of correlation between
pairs of homologous and non-homologous limb bones. He finds that the

36 VARIATION AND CORRELATION IN THE CRAYFISH.

femur and humerus are "distinctly more closely correlated" than are the
femur and the radius. The tibia and radius appear to be "slightly more
correlated" than the tibia and humerus. His general conclusion is that
"serially homologous bones tend to be more closely correlated than non-
homologous bones." Similar results have been obtained by Lewenz and
Whiteley (loc. cit.). In their study of the intercorrelations of the bones
of the hand they found that homologous bones from two digits tend to
be more closely correlated than contiguous bones of the same digit.
Their statement (p. 350) is: "The next highest correlations* are between
lateral and not between longitudinal neighbours, each bone being on the
average more nearly related to the corresponding bone on the next digit,
than to the adjacent bone on the same digit."

It may be said, then, that the evidence at present available indicates
that the morphological relationship implied in the homology of parts is
probably a real factor in influencing the degree of correlation in the
variation of these parts, but that this influence is nowhere a marked one.

PARTIAL CORRELATIONS.

We have so far been discussing the gross correlations between differ-
ent pairs of characters. It is of importance now to examine the "net"
or "partial" correlations of the joints of the legs with one another. The
nature and properties of the coefficients measuring partial correlation have
been fully discussed by Pearson (1902 and earlier papers), and it will not
be necessary here to discuss them in detail. We may, however, note
briefly certain fundamental points in the mathematical theory of multiple
correlation.

Let #,, x2, x3, be any three characters of a population of organisms
varying about their respective means with standard deviations <rn <r2 and
<r3, and organically correlated together to the degree indicated by coeffici-
ents r,2, r,3, r.,3. Then suppose a group of individuals to be selected
from the population with reference to the character xl , so that after the
selection the variability of this character will be that indicated by a stand-
ard deviation sr It has been shown by Pearson (loc. cit.) that in this
selected group of individuals the correlation between x2 and x,3 will be given
by a coefficient

'

* The highest correlations were found to be between corresponding bones of the
right and left hands.

PARTIAL CORRELATIONS.

37

Now suppose that we so select our group of individuals that they shall
all be exactly the same with reference to the character a;,, or, in other words,
eo that after selection there shall be no variation in respect to xr The
standard deviation after selection, s,, will thus of course be zero. Putting
s, =0 in (i) we have at once

**,=

which is the well-known expression for a partial correlation coefficient.
This coefficient measures the correlation between x2 and x3 in a group of
individuals where x1 is constant. If, for example, we let xl denote length
of cephalothorax, x2 length of the great chela, and x3 length of the carpo-
podite of the cheliped, then t.23 measures the correlation between the last
two characters in a group of individuals all having the same length of
cephalothorax.

The partial correlations of the different joints of the legs with each
other were studied in order to get further light on the factors which
influence the degree of the gross correlations. It was decided to deter-
mine the partial correlation between every possible pair of joints avail-
able in our data when the cephalothorax length was made a constant.
The length of the cephalothorax may be taken as an adequate index of
the size of the body, and was on this account chosen as the character to
make constant in the calculations. Calculating from (n) above, and
making cephalothorax length in every case the xl character, we have found
the system of partial correlation coefficients between the different joints
of the legs given in table 18. In the calculations the gross coefficients
which were substituted in equation n were kept to six places of figures.

TABLE 18. — Partial correlation coefficients between the joints of the legs, the cephalo-
thorax length being kept constant.

SEGMENT.

MERIPODITE.

CARPOPODITE.

PHOPODITE.

I.

II.

III.

I.

II.

III.

I.

II.

III.

Meripodite:

1
0.7087
.5968

.5328
.2721
.2761

.4570
.4892
.4969

0.7087
1

.7722

.5793
.4606
.3385

.5669
.7429
.6203

0.5968

.7722
1

.5647
.3501
.3486

.5717
.6184

.6988

0.5328
.5793
.5647

1
.4112
.3312

.7200
.5105
.4443

0.2721
.4606
.3501

.4112

1
.5256

.3580
.3750
.3713

0.2761
.3385
.3486

.3312
.5256
1

.3204
.3127

.3688

0.4570
.5669
.5717

.7200
.3580
.3204

1
.5534

.4605

0.4892
.7429
.6184

.5105
.3750
.3127

.5534
1
.5759

0.4969
.6203
.6988

.4443
.3713

.3688

.4605
.5759
1

II

HI

Carpopodite:

II.

Ill

Propodite:

II.

Ill

38

VARIATION AND CORRELATION IN THE CRAYFISH.

The most striking fact about these coefficients which one notices at
once is that they are uniformly considerably smaller than the correspond-
ing gross correlation coefficients. This means that in a group of indi-
viduals all having the same length of cephalothorax the correlation is
much less close between different segments of the legs than when indi-
viduals of all sizes are considered together. Or, in other words, the result
shows that a considerable portion of the high gross correlations between
joints arises from the fact that both the joints of the given pair are corre-
lated with the length of the cephalothorax. When we get rid of the
effect of these general "size" correlations by making cephalothorax length
a constant we have left the measure of the net relationship between parts.
This net correlation measures the true organic relationship which exists
between a given pair of organs, quite apart from the fact that both the
organs have their dimensions correlated with the size of the body as a
whole. The biological interpretation of the excess of gross over partial
correlation coefficients appears to us to be that it measures the portion
of the total correlation, which is a true growth correlation.

It is of interest to see what proportion the partial bears to the total
correlation in the case of the different legs and segments. This may be
done by expressing the partial coefficient as a percentage of the total.
Doing this for the three possible joint pairs of each leg, we have the result
set forth in table 19.

TABLE 19. — Proportion of net to total correlation in the different legs.

PERCENTAGE o

F NET TO TOTAL

, COEFFICIENT.

Leg I.

Leg II.

Leg III.

Meripodite with carpopodite

55.6

50.7

39.9

Carpopodite with propodite

74.4

41.5

42.1

Meripodite with propodite

48.4

76.6

71.8

Mean

59.5

56.3

51.3

From this table, considering first the means of the columns, we see
that on the average the net correlation forms the largest proportion out
of the total correlation in the case of leg I, less in leg n, and least in leg
in. Or, put in another way, the fact that the joints are correlated with
cephalothorax length measuring size of body accounts for the smallest

PARTIAL CORRELATIONS. 39

part of the observed total correlations between the joints in the case of
leg i, and as we pass to more posterior legs the effect of this factor increases.
From the individual values we begin to get a closer insight into the rela-
tive effects of the different factors influencing the intercorrelations of the
joints of the legs. It is clear that leg I shows distinctly different rela-
tions than do legs II and in. In the case of legs n and in, a very con-
siderable proportion ( >>50 per cent.) of the total correlation of joint pairs
involving the carpopodite as one variable arises from the fact that these
joints are correlated with the length of the cephalothorax. In the joint
pair which does not involve a carpopodite, a very small proportion of the
total correlation is due to this "general size" factor. On the other hand,
leg I shows exactly the opposite relation. There the net relation is least
in the joint pair which does not involve the carpopodite and greatest in
the cases where the carpopodite is included. Leg I follows an entirely
different rule in the correlation of its joints than legs n and in. In
leg i the bulk of the total correlation represents net organic relationship
between the joints, whereas in the other legs a very large portion of the
total correlation arises in an indirect way through the correlation of the
joints with the size of the body as a whole.

Turning again to the values in table 18, we have calculated the mean
values of the net coefficients for the correlations between homologous
joints of the different legs with the following results: Mean net correla-
tion between homologous joints: Meripodites - - 0.6926; carpopodites
= 0.4227; propodites == 0.5299.

For all possible pairs of non-homologous joints of contiguous legs
(12 cases) the mean net coefficient is 0.4789, while for the six cases of
non-homologous joints of non-contiguous legs the mean is 0.4457, or, in
other words, there is no evidence in these correlations of an effect of the
contiguity of metameres. From the means of the net correlations we see
that just as in the case of the total correlations there is clear evidence
that homologous segments tend to the more highly correlated than non-
homologous, but as before the carpopodite correlations form an exception.
These results point to the conclusion that the higher gross correlations of
contiguous parts arise through the growth correlation factor, while for
the higher correlation of homologous parts another explanation must
be sought.

40 VARIATION AND CORRELATION IN THE CRAYFISH.

INDEX CORRELATIONS.

It has been elsewhere pointed out by the writer of this paper (Pearl 1906
and 1907) that it is of considerable theoretical importance to determine for
as many organisms as possible whether there is or is not a sensible correla-
tion between the proportionality of the parts in a differentiated system of
organs and the absolute size of the system. An essential part of Driesch's
so-called "first proof of the autonomy of vital phenomena" (cf. Driesch
1901) depends on the assumption that such a correlation does not exist,
but that, on the other hand, proportionality of parts and absolute size are
quite independent in the organism. Fortunately the matter is one which
can be quantitatively tested and a definite answer reached by direct appeal
to the facts. The proportionality of a series of parts can always be measured
by forming from the absolute dimensions of these parts a series of indices
which will give the percentage which the size of a given part is of the
size of some other part or of some dimension measuring the size of the
whole organism. Having determined these indices, in order to answer
our question we have merely to calculate by now well-known mathematical
methods the correlation between a given index and any chosen measure
of the absolute size of the organism. The mathematical theory of index
correlations was first investigated by Pearson (1897) and Galton (1897).
A discussion and illustration of the meaning of the formulae has been
given by the present writer in another paper (Pearl 1907) and need not
be repeated here. We need merely to note that if xl /xs be any index,
then the gross correlation between this index and the absolute dimension

a?3 is given by the expression

ri3 vi —v3

P = — — — — — — — — — — — — — — — — — — -~

1/Vi2 + v32 — 2ria Vi v3

and the spurious correlation between the two characters which exists
when all organic correlation between x} and a?3 is destroyed is given by

_

"

where rn is the coefficient of correlation between xl and x3t and v1 and vf
are the coefficients of variation of these two characters.

In the crayfish appendages we have a system of parts in which definite
proportions are maintained with a very high degree of constancy. This
is an obvious fact from even cursory observation, and quantitative proof
of it is given in table 6 above. It seems an especially suitable object on
which to test our question as to whether or not these proportions are cor-
related with the absolute size of the body. To get at the question in a

INDEX CORRELATIONS.

41

practical way we have determined the correlations between (a) the index
formed by dividing the length of a joint of a given leg by the length of
the cephalothorax, and (6) the latter dimension, for each segment of the
three legs studied. In other words, we have put for x3 length of cepha-
lothorax and for xl the length of each joint of the legs taken individually.
The results are shown in table 20.

TABLE 20. — Index correlations in the crayfish.

CORRELATED CHARACTERS.

GROSS
CORRELATION.

SPURIOUS
COBRELATION.

Head — Cephalothorax index, with cephalothorax

0.2049

—0.6529

Mer. I. Cephalothorax index, with cephalothorax. .

.4213

- .6418

Mer. II. Cephalothorax index, with cephalothorax. .

.1965

- .6514

Mer. III. Cephalothorax index, with cephalothorax. .

.1347

- .6768

Carp. I. Cephalothorax index, with cephalothorax. .

.4879

- .6108

Carp. II. Cephalothorax index, with cephalothorax. .

.2011

- .6262

Carp. III. Cephalothorax index, with cephalothorax..

.0473

- .6408

Prop. I. Cephalothorax index, with cephalothorax. .

.6753

- .5207

Prop. II. Cephalothorax index, with cephalothorax. .

.1397

- .6734

Prop. III. Cephalothorax index, with cephalothorax. .

.0617

- .9824

From this table we see that in every case the gross correlation between
the index and the absolute length of the cephalothorax is positive. The
values of the coefficients vary greatly for the different indices. As is to
be expected for arithmetical reasons, the spurious correlation is in all cases
negative (cf. formula for pv above). It is to be noted that the value of
the spurious correlation runs much more closely the same for all the
indices than does the gross correlation. This is, of course, what we should
expect a priori, from the very fact that we are dealing with a spurious
correlation ; that is, one whose origin is arithmetic rather than organic.

The fact that, as shown in this table, the observed correlations between
index and absolute size all have the positive sign, while the spurious cor-
relations all have the negative sign, demonstrates the essential point at
issue, namely, that there is a true organic correlation between the indices
(measuring the proportions of the different parts) and the absolute size of
the body as measured by the length of the cephalothorax. There is no
possibility in this case of the argument being made that the interpretation
of the results is doubtful, because we can not be absolutely certain as to
what portion of the observed correlation is organic and what portion
spurious. Here the spurious correlation must in the nature of the case be
negative. But the actually observed gross correlations are in every case
positive. In other words, the tendency toward a negative correlation

42

VARIATION AND CORRELATION IN THE CRAYFISH.

indicated by the high spurious coefficients has been overcome and there
is in every case an additional balance on the positive side. This shifting
of the correlation from the negative direction of the spurious through zero
to the positive side, as observed in the gross coefficients, can only be due
to the influence of a positive organic correlation between index and abso-
lute size. It is immaterial for the essential problem what the exact amount
of this correlation is, or how it shall be measured.* The figures show
beyond any doubt (a) that an organic correlation between the indices
and the absolute dimension exists; (6) that this correlation is in the posi-
tive direction; and (c) that while it varies for the different characters it
is generally high, and certainly to be regarded as significant.

The relation of gross and spurious coefficients in these index correla-
tions may be illustrated by fig. 8.

FIG. 8. — To illustrate the relation of gross and spurious coefficients
in index correlation. (Explanation in text.)

In this diagram the semicircular arc is taken to represent the whole
possible range of values of a correlation coefficient from - -1 through 0
to +1. The positive correlations are on the right of 0 and the negative
011 the left, the directions being indicated by the outside arrows. The
heavy line at A we may consider to represent the actual value of the
correlation between an index, say, carpopodite of leg u/cephalothorax,
and the length of the cephalothorax which one would find if he were
to calculate the index for each individual from a correlation table, and

* Pearson has suggested the expression p — p0 as a measure of the intensity of the
"net" organic correlation, in the case of indices. It is obviously defective, in that it is
not necessarily limited in the values it may take to the 0 and ± 1 of a true coefficient
of correlation. There can be no doubt, though, that in a general way it measures the
"shift" of the correlation. This point has been more fully discussed and illustrated
in the papers of the present writer referred to above (p. 40).

INDEX CORRELATIONS. 43

evaluate the coefficient in the usual way. Now, if in calculating the
indices the carpopodite lengths and cephalothorax lengths were put
together in pairs quite at random (say the carpopodite length of indi-
vidual X was divided by the cephalothorax length of another individual,
F, instead of by its own), instead of as they actually occur, there would
still be a correlation between these indices and the length of the cepha-
lothorax. This correlation we know would be negative, and its amount
may be indicated by the heavy line at B. Now, if there were no organic
correlation between our two characters, index and absolute dimensions,
obviously we should expect that the observed correlation would be that
which arises for arithmetic reasons solely. In other words, we should
expect observed and spurious to be equal, or, on the diagram, A and B
to coincide. But the two are not equal. From the spurious or arithmetic
correlation value at B there has been a shift in the direction of the inside
dotted arrow, through "no correlation" at zero to a positive correlation
measured by the observed coefficient at A. This shift is the result of
the organic correlation between the index and the absolute dimension.
A study of the values of table 20 in connection with the diagram will,
we believe, convince even the non-mathematical reader of the reality of
the result that the proportions of the crayfish body are not independent
of its absolute size, but that, instead, the two things are correlated together
to a definite and significant degree.

In order to bring out in another way the generality of this result we
have resorted to a still different method. We have directly determined the
correlation between an index and an absolute dimension which does not
enter as one of the factors in the index. In this case there is no spurious
correlation; the total observed relationship is organic in origin, just as
truly as the correlation between two absolute dimensions is organic. On
account of the considerable arithmetical labor involved we have worked
out completely only one of these cases, but it will serve to demonstrate
the point made. Other examples were carried far enough to show that
essentially the same results would be obtained with them, the differences
being only in the numerical values of the coefficients. In order to make
the tests as fair as possible the characters which were to go to form the
indices were chosen at random. The first pair of characters which were
drawn were (a) meripodite of leg I, and (6) propodite of leg I. The
quotient of a/b was directly calculated for each individual and entered
on the record cards. Then a correlation table was formed between these

44

VARIATION AND CORRELATION IN THE CRAYFISH.

index values as one variable, and the length of the cephalothorax as the
other variable. Now, it is obvious that since cephalothorax does not enter
into the index fraction, there can be no spurious correlation here. The
resulting table is table 21.

TABLE 21. — Correlation surface for the correlation between the index
Meripodite I/Propodite I, and the length of the cephalothorax.

INDEX IN

PEE CENT.

LENGTH OF CEPHALOTHOKAX.

TOTAL

C>J

CJ

N

M

03

N

CVJ

M

«

eq

PJ

C3

N

N

M

IN

C\l

M

a

o

-u
S3

—
CJ

o

£

OJ

M
(M

O

8

O

s

o

S

o

8

o

t—
N

O

CO

CVI

O

8

O

S

o

^H
«O

O

*a
e<i

(M

O3

O

CO
CO

o

•^<

CO

O

1C

CO

o

l§

O

t^

CO

0

M

N

M

N

<M

fl

C<I

N

M

M

OJ

w

M

w

N

oa

1H

3

i— t
C<I

CM
CJ

to

N

•»*
CM

8

%

t—

N

8

S

8

rH
CO

S

S?

•^»

co

c^

S

41 to 43

1

'i

8
13
9
6
4

'3

'i

14
10
3
2
2

'S

10
3

'3
4
3
3

'2

7
4
1

1
5
4
1
1

1

2
4
2

1

1
1
1
1

1

1

•

i'

5
16
44
74
58
49
21
9
5
1
1

283

43 to 45

45 to 47

1

'i

5
9
3

1

8
11
10

1
>2

1

o

17
12
11
5
2

47 to 49

49 to 51

1
4
3

1

5
2

1

51 to 53

1

53 to 55. ..

'i

1

55 to 57

1

57 to 59

1

59 to 61

1

61 to 63

1

Total ....

1

8

11

18

35

50

44

38

19

15

14

12

11

4

2

i

This table shows at once that there is a distinct correlation between
the two variables, and that this correlation is negative. Calculating from
the frequency distribution given by this table and using Sheppard's cor-
rection of the second moment, we find the following values for the mean
and variability of the index:

Mean index = 49.428 ± 0.133
Standard deviation of index = 3.325 ± .094

For the correlation between the index and absolute dimensions, length

of cephalothorax we find

r - - 0.4621 ± 0.0397

This is clearly a sensible value in comparison with its probable error,
and we must conclude, since there is no possibility here of any spurious or
"arithmetic" correlation, that there is a real organic correlation between
the index formed by dividing the length of the meripodite of the cheliped
by the length of the propodite of the same leg, and the absolute size of
the body as measured by the length of the cephalothorax. In other
words, the proportion existing between the length of the meripodite and

INDEX CORRELATIONS. 45

that of the propodite of leg I is not the same on the average in large
crayfish that it is in small. On the contrary, the proportion of these
two joints relative to each other, changes in a definite and orderly man-
ner as we pass from small to large individuals. By taking the indices
formed by the lengths of other joints of the legs we should reach the
same conclusion from them, the numerical values of the coefficients differ-
ing, of course, in each different case.

An essential part of Driesch's vitalistic argument appears to depend
on the assumption that the proportionality of the parts in a differentiated
system is independent of the size of the system. As a result of quan-
titative studies of the proportionality of various parts and characters of
the body it has been shown that the proportions and absolute size are not
independent, but instead are correlated to a sensible degree in the follow-
ing organisms: Chilomonas and Paramecium (Pearl 1906 and 1907);
the crayfish (the present paper) ; in the aphid Hyalopterus trirhodns
(Warren 1902); and in the case of various proportions of the human skull
by Fawcett (1902) and Macdonell (1904). The writer has in hand unpub-
lished data showing the same thing for several other organisms. It is
not our purpose to enter here upon a theoretical discussion of Driesch's
"first proof," as the writer's position has already been set forth elsewhere
(loc. cit. ). Our present aim is merely to give in detail the additional
evidence afforded by this study of the crayfish on the point made in the
earlier paper. Putting all the evidence together, it would appear that
the assumption that proportionality is independent of absolute size in the
organism is not substantiated when an exact quantitative study of the
facts is made. Only by such quantitative study can it be determined
with any degree of precision whether or not there is a definite association
between two varying phenomena.

46 VARIATION AND CORRELATION IN THE CRAYFISH.

SUMMARY OF RESULTS AND CONCLUSIONS.

This study, dealing with eleven characters of the body and appendages
of the crayfish, had for its primary purpose the determination of the rela-
tion of variation and correlation to the morphological factors, differentia-
tion and homology. The results and the conclusions drawn therefrom
may be summarily stated as follows:

(1) Variation in all the characters studied is skew rather than sym-
metrical in its distribution. The skewness is in all cases positive, or the
mean lies above the mode. In respect to the degree of flat-toppedness or
kurtosis the variation curves all deviate from the mesokurtic condition of
the normal curve. In general, we conclude that the variation in the char-
acters of the crayfish studied can not possibly be adequately described by
the normal or Gaussian curve of errors.

(2) The correlations between the different characters studied are gen-
erally of an unusually high order of magnitude. The coefficients are in
general of about the same magnitude as those which have been found for
the correlation between bilaterally homologous organs in other animals.
The regressions are linear throughout.

(3) It is found that the cheliped, which is the most differentiated leg,
is more variable in all the joints studied than is either the first or the
second ambulatory appendage. This result is obtained whether we measure
the variation in the absolute size of the organs or in their relative propor-
tions when referred to some other dimension of the body as a standard
base.

(4) The most variable and the most differentiated and specialized
single part of all those studied is the great chela.

(5) The frequency distributions for the different joints of the cheliped
have, on the average, the greatest skewness of any of the characters
studied. Degree of skewness and degree of relative variability appear to
run parallel in the variation of the characters we have considered.

(6) The correlation between the homologous segments of two legs is
higher when these legs are contiguous than when they are separated by
an intervening leg. In so far, the crayfish furnishes evidence in favor of
a "rule of neighborhood" in correlation such as has been found by
Lewenz and Whiteley in the correlations of the bones of the human hand.

(7) When the correlations of the non-homologous joints of the dif-
ferent legs are considered such a "rule of neighborhood'1 is not found to
hold uniformly.

SUMMARY AND CONCLUSIONS. 47

(8) The evidence as a whole points to the conclusion that homology
is a factor of relatively very slight importance in determining degree of
correlation between parts. In the case of certain joints of the legs (the
meripodites) homologous joint pairs are significantly more highly corre-
lated together than are non-homologous joints. Such a relation does not
hold uniformly for all joints, however, and, furthermore, in the cases where
it does obtain the differences between homologous and non-homologous
joint pairs in respect to degree of intercorrelation are absolutely very
small. The results indicate that, as compared with physiological factors,
morphological relationship is, for practical purposes, a factor of negligible
significance in influencing degree of correlation between parts.

(9) A study of the partial correlations between the joints of the legs
when the cephalothorax length is kept constant leads to the conclusion
that a very considerable part of the gross observed correlations between
the different segments of the legs arises through the high correlation
between the size of these segments and the size of the body as a whole.
In other words, it appears that the degree of gross correlation between
parts in the crayfish is in each case the resultant of two sets of influences.
There is first a general growth correlation factor which accounts for a
considerable part, but not all, of the observed gross correlation. Besides
this growth correlation sensu strictu, there is left a portion of the gross
correlation, to account for which other physiological factors must be
adduced. In the analysis of these factors by experimental investigation
lies the hope of progress in the problem of the origin of organic correlation.

(10) The data show that in the crayfish there is in general a substan-
tial degree of correlation between the proportionality of the parts and the
absolute size of the organism. We are thus able to add one more to the
list of organisms in which this relationship has been shown by quantita-
tive methods to be true. The bearing of this result on certain of Driesch's
theoretical deductions regarding a vitalistic hypothesis is discussed.

48

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLES OF MEASUREMENTS.

The fifty-five correlation tables which furnish the data for this paper are given in
the present section. In each case the measurements are given in millimeters. In
order to save space a special rubric is not given for each table. Instead, on the top
and left side of each table are stated the two characters whose correlation is given by
the table. The plan according to which the tables have been arranged will be self-
evident on inspection.

TABLE 22.

LENGTH

OF

MEBIPODITE.

CEPHALOTHOHAX LENGTH.

CM

CM

CM

CM

CM

CO

CM

CM

CM

CM

CM

M

CM

CM

CM

CM

CM

CM

"3

43

&

S

3

CM

i- (
W

O

^»

CM

M
M

3

M

§3

o

•M
CM

•*

e*j

3

N

in
CM

O
-M

CM

s

o
-*j

•N

t-

CM

3

CM

a

3

w

8
O

8

o

i— t
CO

O

8

0

CO
CO

O

-H*
CO

o

ci?

o

8

3

CM

s

o
-*>

CM

CM

N

w

M

CSI

CM

CM

0»

i- •(

8

T-t

CM

13

CO
CO

3

O

CM

§3

c-
CM

oo

CO

3

g

3

=0

CO

CO

•***
CO

CO

to

CO

LEG I.
6 9 to 74

1

5
1
1

1

13
9
25
39
55
46
26
27
9
12
12
8

7 4 to 79

1

3

7
1

2
1
14

1

2

7 9 to 84

8 4 to 89

q

8 9 to 94

20
4

18
24

8

9 4 to 99

22
17
4

5

18
13
2

9 9 to 10 4

3

7
8

1

10 4 to 10 9 . . ..

2
9
4

10 9 to 11 4

8
3
2
1

11 4 to 11 9

1

7
4

11 9 to 12 4 ...

3
6

2

'4

2

12 4 to 12 9

1

••

12 9 to 13 4

13 4 to 13 9

13 9 to 14 4

14 4 to 14 9

1

1

Total

1

8

11

18

35

50

44

38

19

15

14

12

11

4

2

1

283

LEG II.

5.5 to 5.9

2
5
1

2
11
18
35
45
49
48
22
17
13
13
5
4

5 9 to 63

1

2
9

2

4
12

1

•j

16
15
1

6 3 to 6 7

9

6.7 to 7 1

7
23
18
2

7.1 to 7.5

6
19
14
1
1
1

io

23
5

1
1

7
8
2

7 5 to 79

7 9 to 83.

2
6

7

8 3 to 8 7

2
7
4
1

8 7 to 91

9 1 to 95

6
5
1

2
6
2

1

'i
1

2

'i
1

9 5 to 9 9

9.9 to 10 3. .

10 3 to 10 7

10 7 to 11.1

11 1 to 11 5

1

1

Total

1

8

11

18

35

50

44

38

19

15

14

12

11

4

2

1

283

LEG III.
6.3 to 6.7..

1

4
3

1
9
18
25
37
48
45
41
18
12
9
12
4
3

6.7 to 7.1

1

3

7
1

'e

10
2

1
2
12
14

6

7.1 to 7.5

7.5 to 7.9

2
16
25
6
1

7.9 to 8.3

4
13
22
~4

'i
16
17
1

1

8.3 to 8.7

8.7 to 9.1

1
14
3

9 1 to 95

4
8
3

1
5
4

3

1

9.5 to 9.9

1
3
4
4

9 9 to 10 3

1

1
2
5

•>

2

9

10.3 to 10.7

10.7 to 11 1

11 1 to 11 5

11 5 to 11 9

1

2

11 9 to 12 3

12 3 to 12 7

1

1

Total . . .

1

8

11

18

35

50

44

38

19

15

14

12

11

t

2

1

283

TABLES OF MEASUREMENTS.

49

TABLE 22. — Continued.

CEPHALOTHOEAX LENGTH.

LENGTH

OF

CAEPOPODITE.

Cl

CM

N

eg

CM

N

CM

00

CM

N

CM

CM

CM

CM

CM

CM

CM

7M

0
CM

o

4a
N

oi

T-*

*H
CO

a

w

N
c*J

s

CM

CO
CM

O
•M

CM

S

0
-M

CM

WO

TJ

O

-*J

CM

tO
CM

O

-4J
CM

S

3

•M

8

0

-4-»

w

O>
CM

O

•*3

CM

8
a

CM

i— 1
CO

5

CM

8

O

S?
o

3

o

£

O

c?
O

r-

co

O

CM

CM

CM

CM

CM

CM

43

0

H

8

S

8

8

-t
CM

ITS

CM

<O
iM

s

8

?;

8

»— 1
CO

&

8

••**
CO

C§

8

LEG I.

4 7 to 51

1
'5

5

1

5
12
19
•10
49
46
36
20
22
11
12
6
3

5 1 to 55

1

2

4
2

'3
9
6

2

5 5 to 59

5 9 to 63

g

6 3 to 67

18
11
1

14

23
11
2

2
12
21

7
1

1

6.7 to 71

2
11
20
4
1

1
1

6
6
5

7 1 to 75

1

1

5

8

7 5 to 79

7 9 to 83

3

6
3

2

i

8 3 to 87

1

8 7 to 9 1

3
6

2

4
3
3

1

1

'2

'i
i

••

9 . 1 to 95

9.5 to 99

9 9 to 10 3

10 3 to 10 7

10 7 to 11 1 ..

11 . 1 to 11 5

i

1

Total

1

8

11

18

35

50

44

38

19

15

14

12

11

4

2

••

1

283

LEG II.
3 4 to 36

3
4
1

1
1
5
4

4

7
20
17
31
37
37
40
27
12
16
10
S
7
6
1
2

3 6 to 38

1
6
5
4
1

1

6
6
11
9
2

3 8 to 40

1

1

4 0 to 42

9

4 2 to 44

13
16
17
2
1

2
9
10
13
6

1
1

7
18
8
2

4 4 to 46

1

1
4
7
5
1

4 6 to 48

4 8 to 50

1

2
4
2
5
2

5 0 to 52

1

1

7
3
2

5 2 to 54

1
3

'i

6

1

1

• •

••

*» 4 to 56

5 6 to 58

1

3

4

'i

1

i

'i
i

••

5 8 to 60

6 0 to 6 2

6 2 to 64 ...

6 4 to 66

1

6 6 to 68

i

l

6 8 to 70

7 0 to 72

1

1

Total

l

1

8

11

18

35

50

44

38

19

15

14

12

11

4

2

283

LEG III.

4 2 to 45

1

2
5

1
1

5
3
1

2
5
14
25
40
42
58
33
20
10
14
12
5

4 5 to 48

1

2
9

7

1
1
9
14

7
2

4 8 to 51

1
3
14
19
13

5 1 to 54

-3
13
19

7

2
18
12
6

'i

1
3
10
3
1

1

5 4 to 57

5 7 to 60

6 0 to 63

1
4

3
3
2

2

6 3 to 66

6 6 to 69

6
3
1
2

2
1

5
3
1

6 9 to 72

1

1
5
4
1

i

i

2

'i
i

••

7.2 to 75

7 5 to 7.8

1

••

••

7 8 to 8 1

8 1 to 84

8 4 to 87

1

1

8 7 to 90

9 0 to 93

9 3 to 96

1

1
1

9 6 to 99

1

Total . ..

4

1

8

11

18

35

50

44

38

19

15

14

12

11

2

1

283

50 VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 22. — Continued.

CEPHALOTHORAX LENGTH

LENGTH

OF

PROPODITE

N

<M

w

N

N

N

N

CM

CM

(M

CM

CM

N

CM

CM

CM

CM

CM

S

o

43
CM

CM

2

CM

CM

CM

a

CM

CO
N

O

-*
CM

O

Iffl
CM

O

to

CM

O

t—
CM

O

8

o

8

o

3
o

TH

CO

0

8

O

S3

o

3
o

£
o

8
o

t-

CO

0

CM

CM

CM

CM

M

eg

M

CM

CM

fa

CM

CM

CM

CM

CM

4-»

O

H

05

TH

s

r-t
CM

CM
CM

m

CM

eS

3

s

§3

OO
CM

o>
CM

8

rH
CO

CM
CO

8

3

3

•-D
CO

LEG I.
12.1 to 13.1

1
1

1
4

6
13
19
24
31
37
31
29
21
17
12
5
9
6
6
4
3
1

13.1 to 14.1

2
3
3

1

2
6
2

14.1 to 15.1

1

15.1 to 16.1

3

11
3

1

1

3
12
13

4

16.1 to 17.1

2
6
13
21
5

9

1

2
5
6
16
10
2

1
1

17.1 to 18.1

i

6
9
13

7

0

1

IX. 1 to 19.1

1

19.1 to 20.1

20.1 to 21.1

'3
10
3
9

1
1

1

6
3
2

21.1 to 22.1

22.1 to 23.1

'4
5
1
2
2

1

23.1 to 24.1

1

24.1 to 25.1

1

25.1 to 26.1

1

3

3
2
2

1

3
1
3
2

'i
i
'2

-•

26.1 to 27.1

27.1 to 28.1

28 1 to 29.1

29.1 to 30.1

i
i

••

30.1 to 31.1

31.1 to 32.1

32 1 to 33.1

33 1 to 34.1

1

1

Total

15

1

8

11

18

35

50

44

38

19

14

12

11

4

2

••

1

283

LEG II.
4.9 to 5.2

1
2
4
1

1

6
11
14
24
44
46
41
30

*>9

12
12
11

5
2

1

5.2 to 5.5

1

1
5
3
2

2

'7

8

5.5 to 5.8

2

2

8
18
5

5 8 to 6.1

1
4

19
22
~4

6 1 to 6.4

2
7
13
16
4
2

6.4 to 6.7

6 7 to 7.0

1

4
20
10
4

1
1
11
6

7.0 to 7.3

7.3 to 7.6

5
5

5

7.6 to 7.9

4

5
5

1
1
4
5
1

7 9 to 8.2

1
2
5
3

i
i

'i
i

i
i

••

••

8.2 to 8.5

8 5 to 88

8 8 to 9.1

9 1 to 94

9.4 to 9.7

9 7 to 10.0

10.0 to 10.3

1

1

Total

1

8

11

18

35

50

44

38

19

15

14

12

11

4

2

••

1

283

LEG III.

6.6 to 7.O..

1
4
3

1
12
16
28
38
48
45
35
17
13
12
9
6
2

7.0 to 7.4.

1

4

6
1

1
4
12
1

2
T

7.4 to 7.8

7.8 to 8.2

11

14
5

3
16
25
5

1

1
3
15
19

5

1

8.2 to 8.6

3
3
18
12
2

1

8.6 to 9.0

9.0 to 9.4

2
12
4

1
4
6
4

9.4 to 98

1

3
6
4

9.8 to 10.2

1

2
5
3
1

10 2 to 10.6

1
3
4
3

2
'2

'2

• •

• •

10.6 to 11.0

11.0 to 11.4

11 4 to 11 8

11.8 to 12.2

12.2 to 12.6

12.6 to 13.0

1

1

Total

1

8

11

18

35

50

44

38

19

15

14

12

11

4

2

••

1

283

TABLES OF MEASUREMENTS.

51

TABLE 22. — Continued.

HEAD
BREADTH.

CEPHALOTHORAX LENGTH.

M

CM

CM

C-l

CM

CM

CM

eg

M

CJ

CM

c->

eg

w

CM

CM

SM

CM

O
«M

O

v-4
M

O

§3

o

co

CM
O

•**
CM

o

£

O

5D
CM

B
CO

t—
CM

O

OO

w

O

a

o

3

o

CO

o

CM

CO

O

??
O

•^
CO

o

s?

o

8

o

s

0

CM

M

CM

CM

CM

(N

CM

ff-i

w

C*l

CM

CM

CM

CM

CM

CM

CM

.p

o
H

O3

TH

8

*H
CM

£

CO
(M

•*
CM

in

CM

CO
cq

F*
M

s

S

9

iH

m

CM

co

S?

3

s

§8

5.3 to 5.8

1

1

5.8 to 6.3

6.3 to 6.8

6.8 to 7.3

7.3 to 7.8

1

3
4

1
6
11
22
43
47
53
34
12
16
16
10
8
2

7.8 to 8.3

3
6
2

8.3 to 8.8

1

8.8 to 9.3

15
2

1

5
24
6

9.3 to 9.8

15
23
10
2

2
15
21
5

9.8 to 10.3

2
17
18

10.3 to 10.8

4
7
5
3

'i

7
6

1

'i

1

10.8 to 11.3

11.3 to 11.8

11.8 to 12.3

1

••

5
6
2

1

6
3
1

12.3 to 12.8...

3

4
4

'i
i

'2

••

12.8 to 13.3

13.3 to 13.8

13.8 to 14.3

14.3 to 14.8

14.8 to 15.3

l

1

Total

1

8

11

18

35

50

44

38

19

15

14

12

11

2

2

l

283

52 VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 23.

HEAD BREADTH.

LENGTH

OF

MBRIPO-

DITE.

00

CO

00

CO

00

CO

00

CO

00

CO

oo

CO

00

CO

00

CO

00

CO

oo

CO

in

O
4a

co

CO

O

-*j

oo

CD

3

CO

t-

o

43
oo

r~
O
CO

oo

a

00

00
O

05
O

OJ
0

0

O

»H

t-l

M

N

CO

CO

•f

••*

in

O

O

O

O

O

O

O

O

O

O

0

CO

00

CO

OO

CO

OO

CO

oo

CO

00

CO

00

CO

oo

3

+a
1

1

13
9

25
39
55
46
26
27
9
12
12
8

H3

\n

CO

CO

t~

t—

00

OO

o>

O5

O

O

^H

TH

N

1M

CO

1-1

CO

>-l

•«*

»-l

-^

tH

LEG I.
6.9 to 7 4

1

7 4 to 79

2
2
2

6

5

3
2
15

2

2

7 9 to 84

8.4 to 8.9
8 9 to 9.4

7
22
12

1

15
23
6
2

9 4 to 99

17

23
9
3

3
16
12
3

9.9 to 10.4
10 4 to 10 9

1

2
8
2

1

9
4
1
1

10.9 to 11.4

4
2

8
2

i
'e

3

'2

3
3

'2

••

11 4 to 11 9

11.9 to 12.4

1

12 4 to 12 9

129 to 13 4

13 4 to 13 9

13.9 to 14.4
14 4 to 14.9

1

1

283

Total. ..

LEG II.

5.5 to 5.9
5.9 to 6.3
6 3 to 67

1

1

6

11

22

43

47

53

34

12

16

16

10

8

2

• •

1

«>

2
11

18
35

1

'4

7
4

2
6
13
1

1

1
18
19
3
I

2
4
19

18
4

1

6 7 to 71

7.1 to 7.5

5

18
22
6

1
10
17
4
2

45
49
48
22
17
13
13

K

i

4

7 5 to 79

7.9 to 8.3
8 3 to 87

1

2

8
2

1

4
9
2

8.7 to 9.1

3
9
4

i
i

5
3

1

3

1

3

i
1

9.1 to 9.5
9 5 to 9.9

1

••

9.9 to 10.3

10 3 to 10 7

10 7 to 11 1

11 1 to 11.5

1

1

283

Total...

LEG III.

6 3 to 67

1

1

6

11

22

43

47

53

34

12

16

16

10

8

2

••

1

1

1
3

1

1

9
18
25
37

6 7 to 71

1

5
6

1

6
12
2

1

1

3
10
19

7
3

7.1 to 7.5

7 5 to 79

2
12

26

7

7 9 to 83

3
9
23
16
1

1
4
12
14
2
1

8.3 to 8.7
8.7 to 9.1

1

48
45
41

18

12
1

12

*
i

9.1 to 9.5
9.5 to 9.9

7

5

4

7
5

3
4
5
4

i

2
5
1
1

'i
i

3
1

2

'2

• •

••

9.9 to 10 3

10.3 to 10.7

1

10.7 to 11.1

11.1 to 11.5

11.5 to 11.9

11 9 to 12 3

12 3 to 12.7

1

1

283

Total . .

1

1

6

11

22

43

47

53

34

12

16

16

10

8

2

1

TABLES OF MEASUREMENTS.

53

TABLE 23. — Continued.

HEAD BEEADTH.

LENGTH

OF

CARPOPO-

DITE.

00

co

oo

co

00

CO

OO

CO

00

CO

00

co

00

03

00

CO

oo

eo

CO

CO

id

3

CO

co
O

-fc3

00

CD

.*->

CO

t-

B

00

t*

o

-M
CO

CO

5

00

CO

o

oi
O

O5
O

o

0

Y— 1

*— 1

M

CM

CO

CO

^*

•**

0

O

O

o

o

O

0

3

00

o

O

O

O

eo

00

co

00

CO

00

CO

00

CO

CO

CO

CO

CO

eg

49

O

E-i

1

5
12
19
40
49
46
36
20
22
11
12
6
3

m

m

CO

co

t-

t*-

00

oo

as

OS

o

o

r-t

i^

SM

IN

1-1

CO
1-H

CO

!-*

-*

*- «

••*

LEG I.

4.7 to 5.1

1
1

5
4

5.1 to 5.5
5 5 to 5.9

1

1

4

1

'3
13
5

2

5.9 to 6.3

1

20
17
3

6 3 to 67

14
21
11
1

1

8
21
19
2

1

1

6 7 to 71

3

7 1 to 7.5

10
13
6
2

7.5 to 7.9
7 9 to 83

1

1

7
4

1

3
10
1
1

2
4

5
4
1

i

2

4
3

2
3
1
2

i
1

8 3 to 87

8 7 to 9.1

9.1 to 9.5

9 5 to 99

9 9 to 10.3

10 3 to 10.7

10 7 to 11.1

11.1 to 11.5

1

1

283

53

Total...

1

••

1

6

11

22

43

47

34

12

16

16

10

8

2

• •

1

LEG II.
3 4 to 36

2
2
1
1

2
2
4
3

4

1

20
17
31

3 6 to 38

1

1

10
5
3

2

1

5
5
17
10
5

3 8 to 40

4 0 to 42

3
6
19
13
5
1

4 2 to 44

4
5
11
17
11
4

1
1

7
13

9
2

4 4 to 46

37

4 6 to 48

1
3

37
40
2'
12
16
10
8

t

4.8 to 5.0
5 0 to 52

1

1

4

2

1
1

2
2
11

1

5 2 to 54

1

•_>

6
2

4
1

1
2
1
2
2
2

'i

3
'3

i
i

5.4 to 5.6

5.6 to 5.8

5 8 to 60

6 0 to 62

6 ° to 64

6 4 to 66

1

6 6 to 68

1

1

6.8 to 7.0

7 0 to 72

1

283

Total...

LEG III.
4.2 to 4.5
4 5 to 48

1

1

6

11

22

43

47

53

34

12

16

16

10

8

2

••

1

1

9

u

3

1
2
5
2

i

1

1
9;

40
42
58
33
20
10
14
12

'3
12
6
1

1
1
9
19
9
3
1

4 8 to 51

1

1
1
10
19
13
1
1

5 1 to 54

'4
9
23

18
3

••

1

5 4 to 57

5 7 to 60

4
15
11
4

6.0 to 6.3
6 3 to 66

1

2
4
3
1

'3
6
4
3

1

6.6 to 6.9
6.9 to 7.2

3
3
4
5

'2
3
3
2

4
2
2

i
l

• •

7.2 to 7.5

7 5 to 7.8

1

7 8 to 81

8 1 to 84

8 4 to 87

1

1

8 7 to 90

9 0 to 93

9 3 to 96

1

'.

9 6 to 99

1

Total . .

1

1

6

11

22

43

47

53

34

12

16

16

10

8

2

1

283

54

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 23. — Continued.

LENGTH

OF

PEOPO-

DITE.

HEAD BBEADTH.

"3

~>

1

00

CO

00

CO

00

CO

00

CO

00

»o

09

CO

00

CO

CO

CO

00

CO

00

CO

U5

o

45
CO

<o

3

00

to
o

-*J

CO

f~

o

-*3
00

t—

3

CO

00

o

-tf
00

00

a

CO

oi
0

4->
00

o>
O

-*3

co

o

0

1—t

1— 1

CM

CM

CO

CO

-#

M*

10

o

-fcj
DO

5

CO

o

-4J

00

2

CO

O

O

o

o

o

3

co

2

00

00

CO

00

CO

CO

10

10

to

to

t»

t—

00

00

OS

OS

0

o

^H

*-«

N

M

CO

CO

•*

•**

LEG I.
12 1 to 13 1

1
1

1

4

6
13
19
24
34
37
31
29
21
17
12
5
9
6
6
4
3
1

13 1 to 14 1

1

1

2
3

1

4
5
1

14 1 to 15 1

15 1 to 16 1

5
12
3
1

1

16 1 to 17 1

4

12
15
10

2
8
13
12
9
3

17 1 to 18 1

'3

7
15
19
6
1
1

1

2
6
6
6
9
3
1

18 1 to 19 1

19.1 to 20.1
20 1 to °1 1

1

1
1
3
5
1
1

21 1 to 2° 1

22 1 to 23.1

2
6
6

2

1

1

3
2

5
3

1

'i

'2

2
3
2

'2

1

2
1

2

'i
1

••

23 1 to 24.1

24 1 to 95 1

25 1 to 26 1

26.1 to 27.1

27.1 to 28.1

28 1 to 29 1

29.1 to 30 1

1

30 1 to 31 1

31 1 to 32 1

32.1 to 3il

33 1 to 34 1

1

1
283

Total...

LEG II.
4.9 to 5 2

1

1

6

11

22

43

47

53

34

12

16

16

10

8

9

1

1

1

6
11
14

24
44
46
41
30
22
12
12
11
5
2
1

5.2 to 5 5

2
3

1

2
5
3
1

2
1
8
9

2

5 5 to 58

2
1
10
19
11

5 8 to 61

'3
20
17
6

1
1
3
12
22
12
1
1

6.1 to 6 4

6 4 to 67

6.7 to 7.0
7.0 to 7 3

1

*>

13
9
6
1

7 3 to 76

5
6
1

4
6
4
2

7.6 to 7.9

1

2
3
5
5
1

'i

3
4
2

i

2
2
1
1
1

'i
1

7.9 to 8 2

8.2 to 8 5

8.5 to 8.8

8.8 to 9.1

9.1 to 9 4

9.4 to 9.7

9 7 to 10 0

10 0 to 10 3

1

1

283

Total...

1

1

••

6

11

22

43

47

53

34

12

16

16

10

8

9

1

LEG III.
6.6 to 7 0

1

1

4

1
12
16

28
38

7 0 to 74

1

6
4
1

2
6
12
1

1

2
2
12
16
11

7.4 to 7 8

7.8 to 8 2

3
14

22
7
1

8.2 to 8.6
8 6 to 9.0

1

5

11

23
13

1
3
14
13

2
1

48
45
35
17
13
12
9
6
2

9.0 to 9.4

'i

7
1

1
4
5

6

9 4 to 98

9.8 to 10.2

3
«>

<5

2
2
4

2

'i

4
1

2

'2

10.2 to 10.6

1

10.6 to 11.0

11.0 to 11.4

i
i

11.4 to 11.8

11.8 to 12.2

12.2 to 12.6

12.6 to 13.0

1

1

Total...

2

1

1

6

11

22

43

47

53

31

12

16

16

10

8

1

283

TABLES OF MEASUREMENTS

TABLK 24.

55

LENGTH OF MERIPODITE.

LENGTH OF MEHIPODITB, LEG I.

•*

Oi

-*

03

•«*

cs

•*

O3

^

O3

•*

en

••*

03

•**

o»

"3

-M

O

H

t-

s

a>

t—

C

00

o

00

o

03
O

O3

O

o

o

*-i

»-l

cj

w

TO

m

««

•***

O

s

•**

o

-^

03

O
-fc?

•^

o

4-.
01

3,

•**

O

-»J

at

O

-4J
•*

O

-t-^

O3

a

-^

•*<

O3

^*

03

•*

O3

ID

t-

t-

00

00

03

O»

o

O

t-(

T— 1

M

CQ

m

CO

«iM

LEG II.
5. 5 to 5.9..

2
7
4

2
11

18
35
45
49
48
22
17
13
13
5
4

5.9to 6.3

1

3

6

6.3to 6.7

6
19

12

24
3

1

4
18
26
5
1

1

6.7to 7.1

7.1 to 7.5

3
17
23
1

1

7.5to 7.9

3
15
7
1

7.9to 8.3

5
11
10
1

8.3 to 8.7

2
5
2

8.7 to 9.1

9.1 to 9.5

7
5

3
6
2
1

9.5 to 9.9

2
3
3

' •

9.9 to 10. 3

10. 3 to 10. 7

10.7 to 11.1

11.1 to 11. 5

1

1
283

Total

1

13

9

25

39

55

46

26

27

9

12

12

8

1

LEG III.
6.3 to 6 7.

1

7
5

1

9
18
25
37
48
45
41
18
12
9
12
4
3

6 7 to 71

1

1
7
1

7 1 to 75

6
16
3

7 5 to 79

7
18
14

1
14

22

16
2

7 9 to 83

2
12
21
11

8 3 to 8 7

8.7 to 9 1

8
16
2

9.1 to 9.5

10
13
4

2
3
3
1

9 5 to 9 9

9.9 to 10.3. .

3

5
4

2
3
6
1

'2
3

* '

10.3 to 10.7

10.7 to 11.1

11 1 to 11.5

11.5 to 11.9 . .

3

11 9 to 12 3

12 3 to 12 7

1

1

Total

1

12

13

9

25

39

55

46

26

27

9

12

8

1

283

56 VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 24. — Continued.

LENGTH OF CARPOPODITE.

LENGTH OF MERIPODITE, LEG I.

•*

O5

•*

o»

-*

05

•^<

O5

•*

05

-*

05

•**

O5

•"*

O5

^

01

43

1

t-

o

49

o>

c-

3

-*

00

o

4>

05

00

o
-u

•<*

05

O
•£

O5

01

o

-u>
»*

o

0

^

^

CM

CM

CO

O5

•«

•^1

2

OS

o

4J

•*

o
-»j

o»

S

•*

O

-M

05

O
-ij

•*

o

•*-•

O5

3

•^

S

05

O

4J

^*

05

tr-

c-

00

00

0>

o>

O

o

I— t

i— 1

CM

CM

n

CO

-*

LEG I.
4. 7 to 5.1.

i

4
5
3

1

5
12

19

40
49
46
36

5.1 to 5.5

1

5.5 to 5.9

5
4

2
11
12

5. 9 to 6.3. .

1

17

19
2

6. 3 to 6.7. .

10
22
21
2

1

7
16
19
3

6.7to 7.1

1

7
10
5
3

7.1 to 7.5

7.5to 7.9

3
10
13
1

2

2
4
1

7.9to 8.3

20
22
11
12
6
3

8.3to 8.7

'S

6
1

2
3
4
3

'i

2
2

3

8. 7 to 9.1. .

9.1 to 95

9.5to 9.9

9 9 to 10 3

10. 3 to 10. 7..

10.7 to 11.1

11.1 to 11. 5

1

1

Total

1

13

9

25

39

55

46

26

27

9

12

12

8

••

••

1

283

LEG II.
3.4 to 3. 6...

3
3
5

1

1
2

3
3

4
7
20
17
31
37
37
40
27
12
16
10
8
7
6
1
2

3.6 to3.8

1

1

10

7
6

1

3.8 to 4.0...

1
2

15
16
5

1

3
7
16
15
9
3

4.0to4.2

1
2
2
14

18
9

4.2 to 4. 4. .

1
2

1

10

7
4

4.4 to 4.6...

4.6 to4.8

1
2

8
5
9
2

1

4.8to 5.O....

1

5.0 to 5. 2...

5. 2 to 5. 4

1
5
2

1

1
2

3
3
2

'i

3
4
3
1

1

i
i

i

3

5.4to5.6

5.6 to5.8

5.8 to 6.0

6.0to6.2

6.2to6.4

6.4 to6.6

1

6.6 to6.8

1

1

6.8 to7.0

7.0to7.2

1

1

Total

1

13

9

25

39

55

46

26

27

9

12

12

8

1

2S3

LEG III.
4.2 to 4.5..,

2
3
6
2

2

5
14
25
40
42
5S
33
20
10
14
12
5

4.5to4.8

2
2

4

1

4. 8 to 5.1

1

4
10
11

'e

14
16
2

1

2
11
16

20
3

1

5.1 to 5. 4...

'2
8
24
11
1

i
«>

7
9
6
1

1

5.4to5.7

5.7to6.0

6.0to6.3

4
10
9

1
2

1

6.3 to6.6
6.6 to6.9

2
3
1
1

1

1
6
4

'i

4
3

1

i
3
4

6.9 to 7.2...

7.2 to7.5

7.5 to 7.8

1

7.8 toS.l. ..

8.1to8.4

8.4 to8.7

1

1

8.7 to9.0

9.0to9.3

9.3 to 9. 6

1

1
1

9.6to9.9

1

Total

1

13

9

25

39

55

46

26

27

9

12

12

8

1

283

TABLES OF MEASUREMENTS.

57

TABLE 25.

LENGTH OF PROPODITE.

LENGTH OF MERIPODITE, LEG I.

•<*

01

•«*

01

•**

OS

•«*

OS

«*

Ol

•*

Ol

-*

Ol

••*

01

"3

4a

£

c-

3

01

r-

O
-fca

•*

CO

o

-*J

O5

00

3
•"*

Ol
O

oi

o

o

o

r*

i— »

CM

M

m

CO

-*

-^

o

o

O

o

o

2
••*

O
-fcj

Ol

3

•"*

2

Ol

5

•«*

01

••*

01

-*

01

•»*

o>

co

t-

t»

CO

00

Ol

01

o

0

1-*

i— i

CM

CM

CO

CO

-#

LEO I.
12 1 to 13 1

1

3
4
4
1

1

4
6
13
19
24
34
37
31
29
21
17
12
5
9
6
6
4
3
1

13.1 to 14.1

1

14 1 to 15 1

2
5
2

15 1 to 16 1

3
13
6
3

1

2
8
16
11
1

16 1 to 17.1

1

9
10
17
11
6

17 1 to 18 1 ....

1

3

6
15
15
6.

18 1 to 19 1

2
2

3

6
10
3

19 1 to 20 1

1
1

2
3
10

8
2

20 1 to 21 1

91 1 to 22 1

22 1 to 23 1

2
3

2

1
1

23 1 to 24 1

'i

4
4
2

1

'i
i

3
2
3
2

1

'i

'i
1

3
1

••

24 1 to 25 1

1

••

25 to 26 1

26 to 27 1

27 to 28 1

•>8 to 29 1

29 to 30 1

30 to 31 1

31 to 32.1

32 to 33 1

33 to 34 1

1

1

Total

1

13

9

25

39

55

46

26

27

9

12

12

8

••

••

1

283

LEG II.
4 9 to 5 2

1

1
6
11
14
24
44
46
41
30
22
12
12
11
5
2
1

5 2 to 55

6
4
2

1

5 5 to 5 8

6
2
1

1

9
12
3

5 8 to 6 1

'5
22
12

1
5
15
21
9
3
1

6 1 to 6 4 .

6 4 to 67

4
12
25
5

6 7 to 70

1

7
11

7

7 0 to 73

7 3 to 7 6

10
9

8

1
4

2
2

7 6 to 7 9

'i

6
5

1
1
3
4
3

i

2
2
2
1

••

. .

7 9 to 8 2

8 2 to 85

8 5 to 88

8 8 to 91

9 1 to 9 4 •

9 4 to 9 7 ....

9 7 to 10 0

10 0 to 10 3

1

1

Total

1

13

9

25

39

55

46

26

27

9

12

12

8

1

283

LEG III.
6 6 to 70

1

7
5

'2
6
1

2
4
16
3

f t

. .

r

. .

1
12
16
28
38
48
45
i5
17
13
12
9
6
2

7 0 to 74

1

7 4 to 7 8

1
7
19
11
1

7 8 to 8 2

4
12
25
12
2

8 2 to 8 6

3
10

26

7

1
2
3
17
3

8 6 to 90

9 0 to 94

2

8
9
8

1
1

4
2

1

9 4 to 9.8

9 8 to 10.2

-2
8
1
1

1
1
3
6

1

'z

4

9

• •

• •

• •

10 2 to 10 6

10 6 to 11 0 . ..

11 0 to 11 4

11 4 toll 8

11 8 to 12 2 ...

12 2 to 12 6

12 6 to 13 0

1

1

Total

1

13

9

25

39

55

46

26

27

9

12

12

8

1

283

58

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 26.

LENGTH OF CARPOPODITE.

LENGTH OF MERIPODITE, LEG n.

o>

ec

t—

1-1

in

OS

TO

t—

T— 1

in

OS

CO

t-

T-H

m

"3

+a
O

H

in
O

49

in

to
O

•*3

O5

to

S

w

i—

2
t*

r-

O

r*

O

00

o

00

O

OS

O

OS
O

OS

$

m

o

1— 1

O

-*J

OS

o

1"H

3

CO

I—I

^•1

5

c-

1—4
rH

O
-u

^H

i— i

in

0)

w

c-

^

in

»o

«D

co

t-

c—

c—

00

00

OS

OS

05

o
1-1

0
TH

1— t

^H

LEG I.
4.7to 5.1

i

i

1

3
5
2

1

5
12
19
40
49
46
36
20
22
11
12
6
3

5.1 to 5.5

1

5
9
1

5. 5 to 5.9

1

8
22
4

5.9to 6.3

6. 3 to 6.7

12

26

7

5
13
22

8

1

6.7to 7.1

6

11

22

7
2

7.1to 7.5

2

3

5
8
6

1
1

2
12
1

7.5to 7.9

7.9to 8.3

2

1
4
5
1

'i

4

5
3

'2

2
1

'i

3

••

••

8.3to 8.7

8.7 to 9.1

9.1 to 9.5

9.5 to 9.9

9.9tol0.3

10.3 to 10. 7

10.7 to 11.1

11.1 to 11.5

1

1

Total

2

11

18

35

45

49

48

22

17

13

13

5

4

1

283

LEG II.
3.4 to3.6

1
1

o

3
4
2

1

3
6
6
1

4
7
20
17
31
37
37
40
27
12
16
10
8
7
6
1
2

3.6 to3.8

3. 8 to 4. 0....

10
6
14
4
1

4.0to4.2

2
14
19
8
1

1

2

10
17
16
3

4. 2 to 4. 4

4.4to4.6

4
10

18
13
2

1

4.6 to4.8

1
3
8
6
2

1

4.8 toS.O

1

1
3
2
9
2

5.0 to 5. 2

5.2to5.4

'3
4

1
3
2

1
1
3
5
2
1

1

i

2
1

i

'2

••

5.4 to 5.6

5.6 to5.8

5.8 to 6.0. .

6.0to6.2

6.2 to6.4

6.4to6.6

1

6.6 to6.8

i

1

6.8 to 7.0...

7.0to7.2

1

1

Total

2

11

18

35

45

49

48

22

17

13

13

5

4

1

283

LEG III.
4.2to4.5

1
1

1
2

4
4

2
5
14

25
40
42

58

4.5 to 4.8

2

8
4
2
1
1

4.8to5.1

1
12
17
4

1
4
15

18
6

5.1 to5.4

'5
12
26
6

1
1
7
19
15
5

5.4to5.7

5.7 to6.0

6.0 to6.3

4
9
8

2
3
4
5
3

6.3to6.6

33
20
10
14
12
5

6.6 to6.9

2

4
5
2

1
1
4

7

-2
3

*>
2

••

6.9 to 7. 2

7.2 to 7.5 .

7. 5 to 7. 8
7.8to8.1

1

8.1 to 8. 4

8.4 to8.7

1

1

8.7 to9.0..

9.0 to 9. 3. . .

1

9.3to9.6

1

1

9.6 to 9 9

• •

1

Total

2

11

18

35

45

49

48

22

17

13

13

5

4

••

1

283

TABLES OF MEASUREMENTS.

59

TABLE 26.— Continued.

LENGTH OF PBOPODITE.

LENGTH OF MEKIPODITE, LEG II.

<3>

>n

c—

T-t

>o

o»

CO

i—

»H

>f>

o>

CO

t—

1-4

o

IO

O

«o
O

CO

o

r-»

O

t-

o

t-

o

CO

O

CO

o

a>

O

O)

O

o>

3

10

o

I-H

o

o
r*

O

>— 1

O

rH
^H

O

o

0)

CO

t—

fH

in

o»

CO

c—

*^

O5

CO

t—

tH

&

£

o

m

<c

«o

t-

c—

lr»

00

CO

0>

0>

o»

o

1-1

o

^

^
^H

LEG i.
12.1 to 13.1..

1

i

4
6
13
19
24
'M

13.1 to 14.1

1
1

3
3
4
1

14.1 to 15.1

2
7
6

15.1 to 16.1

2
11
11

9
2

16.1 to 17.1

1

17.1 to 18.1

11

14
16
3
1

2
7
14
14
9
1

18.1 to 19.1

3
5
12
15
10
2

1

1

19.1 to 20.1

37
31
29
21
17
12
5
9
6
6
4
3
1

20.1 to 21.1

2

21.1 to 22.1

4
6
8
3

22.1 to 23.1

3

5
5
3

1

1
1

2
1
3
4
1

i

4
2
3
3

i

'i
'2
'i

'i
•>

••

••

23 1 to 24 1

24.1 to 25.1

1

25.1 to 26.1

26 1 to 27 1

27.1 to 28.1

28 1 to 29 1

29 1 to 30.1

30 1 to 31.1

31.1 to 32.1

i

32.1 to 33.1

33.1 to 34.1

1

1

Total

17

2

11

18

35

45

49

48

22

13

13

5

4

1

283

LEG II.

4.9 to 5.2

1
3
4
3

1
6
11
14
24
44
46
41
30

92

12
12
11
5
2
1

5.2 to 5.5

2

1

7
4
4
2

5.5 to 5.8

5. 8 to 6.1

6
18
10
1

1

25
19

6 1 to 6.4

2
7
21
18
1

6.4 to 6.7

6.7 to 7.0

5
23
15
5

7 0 to 7 3

7.3to 7.6

12
9
1

2
6
8
1

7 6 to 7.9

2
1

6
4

-2
5
3
3

'4
1

'i

2

1

••

••

7 9 to 8.2

8.2 to 8.5

8 5 to 8 8

8.8 to 9 1.

9 1 to 94

9 4 to 97

9 7 to 10.0

10 0 to 10.3

1

1

Total

2

11

18

35

45

49

48

22

17

13

13

5

4

1

283

LEG III.

6 6 to 70

1
1

1
12
16
28
38
48
45
35
17
13
12
9
6
2

7.0 to 7.4

6
5

4

8
3

1

1
2

19
11
2

7.4 to 7.8

1

5
20
16
3

7.8 to 8.2

1
3

24
15
5
1

8.2 to 8.6

3
6
23
14
2

8.6 to 9.0

9.0 to 9.4

2

1
14
6
1

1

2

6

8

9.4 to 9.8

9.8 to 10. 2

2
3
7
1

'i

5
5
2

'3

2

2
o

10.2 to 10.6

10.6 to 11.0

11.0 to 11.4

11.4 to 11.8

11. 8 to 12. 2

12.2 to 12 6

12.6 to 13.0

1

1
283

Total

2

11

18

35

45

49

48

22

17

13

13

5

4

1

60

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 26. — Continued.

LENGTH OF MEEIPODITE.

LENGTH OF MEEIPODITE, LEG II.

at

09

IT"

1-1

m

os

CO

t-

1-1

in

O3

m

C-

^H

kO

'S

49

1

in
O

49

1C

<£>

3

&

«D

a

co

C-

3
t-

t-
o

-*3

1-1

c-

O
-*j

m

00

$

as

00

O

O5

O

OS

O

O5
O

o
-H

O

O

i-H

s

m

—

i

t*

^H

^

O

-*3

TH

CO

c-

i-<

1C

o>

in

m

eo

«D

c-

t-

t~

eo

oo

OJ

OS

O5

o
i— i

o

^H

—

LEG m.

6.3 to 6.7

1

1

1

9
18
25
37
48
45
41
18
12
9
12
4
3

6.7to 7.1

6
5

2
10
4

7.1 to 7.5

3
18
13
1

7.5 to 7.9

3
22
19
1

7.9to 8.3

2
22
22
3

8.3to 8.7

1
1

5
21
21
1

8.7 to 9.1

9 1 to 9.5

14

8

3
8
5
1

9 5 to 9.9

1

7
3
2

'5

6
1
1

'3

2

i
i

2

..

..

9.9 to 10.3

10.3 to 10. 7

10 7 to 11 1

11.1 to 11.5

11.5 to 11. 9

11 9 to 12.3

12.3 to 12.7

1

1

Total

2

11

18

35

45

49

48

22

17

13

13

5

4

1

283

TABLES OF MEASUREMENTS.

61

TABLE 27.

LENGTH OF CARPOPODITE.

LENGTH OF MERIPODITE, LEG III.

t—

TH

us

OS

CO

t—

i-<

U3

OS

CO

t-

^H

Ut

OS

co

r-

"ffl

49
1

<c

3

CO

r»

5
t—

r-

3

rH

t-

3

US

00

O
-*3

OS

00

O

43

CO

OS

O

-4-»
t-

os
O

-4J
tH

OS

a

m

O

o

»H

1-4

iH

c*i

e-j

O
*^

OS

0
-*i

CO

3

c-

s

^H

0

+3

in

O
-td

OS

O
-*j

CO

to

«c

IT*

t*

t—

oo

00

OS

os

OS

o

0

*—

vH

*-i

e-i

LEG I.

4.7to 5.1

1

1

5
12

19

5.1 to 5.5

1

3
4

1

1

6
8
3

5. 5 to 5.9

2
9
14

5.9 to 6.3

1

16

18
2

6.3 to 6.7

7
21
15
5

40
49
46
36
20
22

11
12
6
3

6.7to 7.1

8
23
12
2

2

5
17
10

7

7 1 to 7.5

1

2
6
9

7 5 to 7.9

7.9 to 8.3

2
4
3
3

8.3 to 8.7

2
5
2

'3

5
4

~2
1

1

'i

2

8.7 to 9.1

9.1 to 9.5

9. 5 to 9.9

9 9 to 10 3

10 3 to 10.7

10.7 to 11.1

11.1 to 11.5

1

1

Total

1

9

18

25

37

48

45

41

18

12

9

12

4

3

1

283

LEG II.

3 4 to 36

1

1
3
4

1

1

4
5
4
3

1

4

7
20
17
31
37
37
40
27
12
16
10
8
7
6
1
2

3 6 to 3.8

3 8 to 4 0

9

7
6
1
1

2
4
11
16
3
1

4 0 to 42

1
9
11
15
11

4.2 to 4 4 ..

1
9
13
15
6
1

1

4 4 to 46

4 6 to 4 8

5
11
15
6
2

4 8 to 5 0

1

1
5
3
7
2

5 0 to 52

1

'5
4

5.2 to 5 4

1
1
1
3
3

1
1

3
3
2
2

'2

1
1

'2

••

••

5.4 to 56

5 6 to 58

5 8 to 6 0 . .

6 0 to 6 2 .

1

1

6 2 to 64

6 4 to 6 6

1

6 6 to 68

1

••

1

6 8 to 70

7 0 to 72

1

1

Total

1

9

18

25

37

48

45

41

18

12

9

12

4

3

••

1

283

LEG III.

1 9 tn 1 ^

1

1
3
4
1

2
5
14
25
40
42
58
33
20
10
14
12
5

4 *> tn 48

2

7
7
2

4 X t (i 1 1

2
11
10
1

'i
20
11
2

1

1
6
20
18

1

5 1 to 54

'i

9
25
9

1

i
1

10
18
10
1

1

5 4 to 57

5 7 to 60

6 0 to 63

2
5

1

6 S to 66

6 6 to 6 9

7
1
1

1
5
4
1

1
3
4
1

'4
7
1

'i
i

2

'i
2

••

6 9 to 7 2

7 2 to 75

7.5to 7.8

1

••

••

••

7 8 to 81

81 tn ft 1

H 4. tn R 1

1

1

93 tn q e.

1

1

1

9.6to 9.9

1

Total

1

9

18

25

37

48

45

41

18

12

9

12

4

3

••

1

283

62

VARIATION AND CORRELATION IN THE CRAYFISH

TABLE 27. — Continued.

LENGTH OF PROPODITE.

LENGTH OF MEEIPODITE, LEG III.

c-

T-*

m

01

CO

t-

TH

in

Oi

CO

t—

t-t

>«

05

CO

f"

i-H

00

*j

i

to

o
-M

co

r—

0
-M

c—

t-

O
+>

1^

I—
O

-M

IO

00

3

as

00

o

•»-»
CO

oi

S

c-

o>

O

o>

o

0

o

l-H

*-i

—

M

(M

o

O

O

O

O

S

01

O
.u

CO

1-4

tfi

O»

CO

c-

fH

in

CO

<0

c-

t-

t-

oo

00

Oi

o>

o>

o

o

*-»

**

T-*

M

LEO I.

12 1 to 13 1

1

i

4
6
13
19
24
34
37
31
29
21
17
12
5
9
6
6
4
3
1

13 1 to 14 1

1

3
3
2

1

14 1 to 15 1

3

8
5
1

15 1 to 16.1

2
11
8
3
1

1

2
10
16
7
1

16 1 to 17 . 1

17 1 to 18 1

4
9
22

"s

5

1

4
5
20
11
2

1

1

18.1 to 19 1

1

2
1
13
18
5
1

1

19 1 to 20

20 1 to 21

1

21 1 to 22

22 1 to 23

1

7
7
1

23 1 to 24

s

3
3
1

9

••

1

••

24 1 to 25

25 1 to 26

1
2
4

'6

..

..

26 1 to 27 1

27 1 to 28 1

28 1 to 29 1

1
1

4
2

1

i

29 to 30 1 .

30 to 31 1

1
1

2

..

31 to32.1

32 to 33 1

33 to 34 1 .

1

1

Total

4

1

9

18

25

37

48

45

41

18

12

9

12

3

••

1

283

LEO II.

4 9 to 5 2

1
3
3
1
1

1

6
11
14
24
44
46
41
30
22
12
12
11
5
2
1

5 2 to 5 5

1

2

7
7
2

5 5 to 5 8

1

5
13
4
2

5.8 to 6.1

1

7
21

8

6 1 to 64

is

23
6
1

1
1
13

24
4
2

6 4 to 6.7

6 7 to 7.0

7.0 to 7.3

11

22
8

7.3 to 7.6

3
10
5

7 6 to 7.9

2
4
4
2

7.9 to 8.2

2
5
1

1

'3
7
2

1

i
i
i

'i
l
l

••

..

8 2 to 8 5

8 5 to 8.8

8.8 to 9.1

9 1 to 9.4

9.4 to 9.7

9 7 to 10.0 .

10.0 to 10.3

1
1

1
283

Total

9

1

18

25

37

48

45

41

18

12

9

12

4

3

••

LEG III.

6 6 to 70

1
6

2

1

12
16
28
38
48
45
35
17
13
12
9
6
2

7 0 to 74

1

5

7 4 to 78

11
2

3
15

7

7 8 to 82

9
21
6
1

2
8
26
12

8 2 to 86

1
16
18
10

1

8 6 to 9 0 . .

9 0 to 9 4 .

13
22
5

1
3

10
4

9 4 to 9.8

9.8 to 10 2

2

7
2

1

10.2 to 10. 6

2
6

1

'4

6

o

'i

3

'i

2

••

10.6 to 11.0

11.0 to 11.4

11.4 to 11. 8

11.8 to 12.2...

12 2 to 12 6

12 6 to 13.0

1
1

1

Total

1

9

18

25

37

48

45

41

18

12

9

12

4

3

283

TABLES OF MEASUREMENTS.

63

TABLE 28.

LENGTH OF
CARPOPODITE.

LENGTH OF CARPOPODITE, LEG I.

i^

in

o>

co

c—

1-1

m

o>

CO

c-

iH

in

o»

CO

t-

1-H

m

ia

o

•*•

c—

IO

S

?H

o

O

CO

o

CD

O

t-
O

t-
O

t~

o

00

o

x>

o

C»

O

o»

3

iH

o>

a

in

o
i^

O

o

^H

O

i"-l

^H

O

*H
1-H

O

if)

a>

CO

c—

iH

in

o»

CO

c—

Oi

CO

t-

^H

0)

*_»

c2

•*

in

in

m

CO

CO

C*

c—

i—

00

00

oi

en

o>

o

*-H

o

.—

V*

f-H

LEG II.
3. 4 to 3.6..

1
2
1
1

3
3
3
2
1

4
7
20
17
31
37
37
40
27
12
16
10
8
7
6
1
2

3.6to 3.8

2
9
6
1

3.8 to 4.0

1

6
4
13
13
4

4.0 to 4.2

3
11
15
14
5
1

1

5

4.2to 4.4

4.4to 4.6

6
13
13
6
1

2
6
16
10
1
1

1

4.6to 4.8

4.8 to 5.0

1

4
7
4
3
1

1

3

4
8
5

5.0to 5.2

5.2to 5.4

2
2
2
3

2

5.4to 5.6

1

2
4
3
2

1

'2
3

'i
i

5.6 to 5.8

5.8 to 6.0

6.0 to 6.2

6.2to 6.4

6.4 to 6 6

1

6.6to 6.8

i

1

6. 8 to 7.0

7.0to 7.2

1

1

Total

1

5

12

19

40

49

46

36

20

22

11

12

6

3

'•

• •

1

283

LEG III.
4.2 to 4.5

1

1
2
1
1

2
5
14

25
40
42

58

4.5 to 4.8

3
5
3
1

4 8 to 5 1

6
11
2

1
6
19
11
2

1
3

15
16
11
2
1

5.1 to 5 4 ..

2
9
25
9

'i

6
16
10
2
1

1

5.4 to 5.7

5.7 to 6.0

6 0 to 63

3

7
8

1

5
7
4
4

6 3 to 6 6 . .

33
20
10
14

12
5

6 6 to 6 9 ... .

2
4
3
2

6.9 to 7.2

1
5
4
2

'2
3
1

'i

2

"

• •

7.2 to 7.5

7.5 to 7.8

1

••

1

7 8 to 81. ...

8 1 to 8 4

8 4 to 8 7

1

1

8.7 to 9.0

9 0 to 9 3

9 3 to 9 6

1

1
1

9.6 to 9 9

1

Total

1

5

12

19

40

49

46

36

20

22

11

12

6

3

1

283

64

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 28. — Continued.

LENGTH OF
PEOPODITE.

LENGTH OF CAEPOPODITE, LEG I.

T-(

in

OS

CO

t—

*-(

m

OS

n

t—

^H

in

OS

CO

t—

rt

in

"i5

43

O

EH

m

3

t-

in

3

!-t

in

2

in

«D

O

to
O

t-

o

c-

O

r-
O

00

O

00

O

OS

O

OS

3

»H

OS

O
-*a

in

O

rH

O
-*A

OS

0

TH

O

+?

00

^H
^H

O

-4J
t~

—
O

*s

^H

OS

CO

r-

TH

in

OS

n

t—

^«

IS

in

in

«o

50

t-

t-

c-

00

00

OS

Oa

os

O
^H

d
^4

i-H
rt

LEG I.
12.1 to 13.1

1

3
1

1

4
6
13
19
24
34

13.1 to 14.1

i

14.1 to 15.1

5
5
2

15.1 to 16.1

8
8
2
1

16.1 to 17.1

8
16
12
4

1

6

20
16
6

17.1 to 18.1

18.1 to 19.1

1

15
17
10
2

19.1 to 20.1

2
6
16
9
2

1

37
31
29
21
17
12
5
9
6
6
4
3
1

20 1 to 21 1

2
3
6
8
1

21 1 to 22.1

22.1 to 23.1

4
6
8
3

1

23 1 to 24 1

1

24.1 to 25.1

1

1

1

3
3
2
1

25 1 to 26 1.

1

5
3
2

'2

3

1

2
1

t t

26 1 to 27 1.

27.1 to 28.1

28.1 to 29.1

29.1 to 30.1

30.1 to 31 1

31.1 to 32.1

32.1 to 33.1

33 1 to 34 1

1

1

Total

i

5

12

19

40

49

46

36

20

22

11

12

6

3

1

283

LEG II.

4.9 to 5.2

1
2
2

1

6
11
14
24
44
46
41
30
22
12
12
11
5
2
1

5.2 to 5.5 .

i

2
6
3

1

1

3
6
9

8.5 to 5.8

5.8 to 6.1

4
12

18
6

1
1
19
20
6
1
1

6.1 to 6.4

1
6
16
18
4
1

6 4 to 6.7

1
3
16
11
4
1

6.7 to 7.0

1

1
9
7
1
1

7.0 to 7 3.

7 3 to 7 6. .

5
9

7

7.6 to 7.9

7.9 to 8.2

2
5
4

1

5
6

'i
i

4

t f

8 2 to 85

8 5 to 8.8. .

8.8 to 9.1

1

9.1 to 9.4

2
1

9.4 to 9.7

9.7 to 10.0

10.0 to 10.3

1

1

Total.

i

5

12

19

40

49

46

36

20

22

11

12

6

3

1

283

LEG III.
6.6 to 7.0...

1

2
8
1

1
12
16
28
38
48
45
35
17
13
12
9
6
2

7.0 to 7.4

i

4
1

4
4
11

1

3
11
14
10
1

7.4 to 7.8

7.8 to 8.2

5
17
17
9
1

8.2 to 8.6

4
17

18
6
1

3
3
14
15
1

8.6 to 9.0...

1
3

8
5

2

1

9.0 to 9.4

9.4 to 9.8

5

9.8 to 10.2

10
6

i

10.2 to 10.6

4
5

2

1
4
5
2

'2

1
3

'i

2

10.6 to 11.0

11.0 to 11.4

11.4 to 11.8

11.8 to 12.2

12.2 to 12.6

12.6 to 13.0

1

1

283

Total

1

5

12

19

40

49

46

36

20

22

11

12

6

3

••

1

TABLES OF MEASUREMENTS.

65

TABLE 29.

LENGTH OF
PEOPODITE.

LENGTH OF CARPOPODITE, LEG II.

<£>

00

o

CM

•»*

co

00

o

<M

•*

CO

00

o

N

•"»>

CO

00

o

N

"cS

"o
H

m

O
45

^J*

CO

O

-*£

•-C'

•^i
o

*a

00

•*

0
-u

o

•*

3.

N

**

S

-t<

•«*

O

-fcS
CD

tti

O
-w

00

10

3

o

10
O

10
O

IO
0

CO

o

CO

O

CO

O

CO

O

CO

O

i—

0

1-

o

N

••*

CO

00

o

Cl

•*

CO

00

o

n

CO

CO

10

ira

IO

o

in

CO

CD

CO

CO

CD

c-

LEG I.

12.1 to 13.1..

1

1
1

3

1

1
4

6

in

19
24
34
37

13.1 to 14.1....
14.1 to 15.1....
15.1 to 16.1....
16.1 to 17.1

1
2
1

1

2
4
7
4
2

1

1

4

6
2
2

'S

5
8
11
o

1

17.1 to 18.1....
18.1 to 19.1....
19.1 to 20.1....

11

10
9
3
2

1

9
11

9
6

1
1

5
13
11

7

'i

2
1
3

8
7
4
2

20.1 to 21
21.1 to 22

1

31
29
21
17
12
5
9
6
6
4
3
1

1

6
2
1

1

22.1 to 23

1

6
5
2

1

23.1 to 24

1
1

1

1

3
2
3
1

1

1

24.1 to 25

25.1 to 26

26.1 to 27

3
2
2

2
2
1
1
1

27.1 to 28

1
1

2
1

28.1 to 29.1....

1
1

1

i

29.1 to 30.1....

30.1 to 31.1....

31.1 to 32.1....

1

32.1 to 33.1....

33.1 to 34.1....

1

1

Total

4

7

20

17

31

37

37

40

27

12

16

10

8

7

6

1

2

1

283

LEG II.
4. 9 to 5.2..

1

2
3

i

1

6
11
14
24
44
46
41
30
22
12
12
11
5
2
1

5. 2 to 5.5....
5. 5 to 5.8....
5. 8 to 6.1....
6.1 to 6.4....

1
3

2
2

7
7
2

'3
4

5
4
1

1

3
9
12
6
1

2
17
13
3

6. 4 to 6.7....
6.7to 7.O....
7.0to 7.3....

8
16
10
2
1

1

8
20
7
2
1

1

7
13
5
1

1

7.3to 7.6....

8
1
2

7.6to 7.9....

2

7
6
2

i

3
1
4
1
1

1

7.9to 8.2....

1
3
3

'i

8.2to 8.5..

3
3
1

-3
2
1

••

"i

8. 5 to 8.8....

1

8.8to 9.1....

9.1 to 9.4....

9.4to 9.7....

9.7 to 10.0...

10.0 to 10. 3..

1

1

Total

4

7

20

17

31

37

37

40

27

12

16

10

8

7

6

1

2

••

1

283

LEG III.
6. 6 to 7.O..

1
3
3

1

12
16
28
38
48
45
35
17
13
12
9
6
2

7.0to 7.4....
7.4to 7.8....
7.8to 8.2....
8.2to 8.6....
8 6 to 90

1
3

6
4

7
3

1
3

9
3

1
2
7
13
6
1

5

1

11
16
3
2

5
15
13
3

1

2

8
18
10

'i

1
3
6
12
4
1

9.0to 9.4..
9 4 to 98

1

1

7
2
1

1
1
5
8
1

1

9.8 to 10 2

1

3

1

10.2 to 10.6

2
3

2

10.6 to 11.0....

3
3
2

4
2
1

1
1
3
1

'i

11.0 to 11. 4....

1

11.4toll.8....

11.8 to 12.2....

12.2 to 12 6

12.6 to 13.0

i

• •

••

1

Total

4

7

20

17

31

37

37

40

27

12

16

10

8

7

6

1

2

1

283

66

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 29. — Continued.

LENGTH OF
CAEPOPODITE.

LENGTH OF CAEPOPODITE, LEG II.

go

90

o

04

•«*

co

00

o

CM

^i

CO

00

o

CM

•^

co

00

o

CM

"3
5

H

co

O
4*

•»*

CO

O

^

to

O
-*J

00

3

o

0

O

O

to

O

L^

O

in

O

C5

O

>n

O

co

O

CO

o

co

O

CO

O

CO

o

tT"

O

C-

0

CM

f

CD

CO

o

CM

•^

CO

00

o

CM

•*

CO

CO

o

«

CO

m

IO

m

in

in

m

CO

CO

CO

CO

CO

t-

LEG III.

4.2 to 4.5
4 5 to 4 8

1

'3

3
1

1
2

3

9
4
1

2

5
14
25
40
42
58
33
20
10
14
12
5

4.8 to 5.1

3

2

7
7

'i

1

5
13

8
3

1

1

12
14
6
2
1

i

3
15
13

5

1

i

28
7
3

5 1 to 5 4

1
1

3
6
12
3
1

5.4 to 5.7
5 7 to 6 0

6 0 to 6 3

'i

7

i

'•>

Q

6
3

5

1

6 3 to 6 6

1

••

6 6 to 6 9

6 9 to 7.2

3
1
4

2
3
3

1

2
2
2

2

2
2

i

7 2 to 7 5

7 5 to 7 8

1

7.8 to 8.1

8 1 to 8 4

8 4 to 8 7

1

1

8 7 to 9 0

9 0 to 9 3

9 3 to 9 6

1

1
1

9 6 to 9 9

1

Total

4

7

20

17

31

37

37

40

27

12

16

10

8

7

6

1

2

1

283

TABLES OF MEASUREMENTS.

67

TABLE 30.

LENGTH OF CAEPOPODITE, LEG III.

LENGTH OF
PEOPODITE.

in

00

TH

•*

c—

o

CO

to

o>

N

in

oo

1-1

•*

t->

o

co

to

01

^

a
4m

O

H

1

4

6
13

19
24
34
37
31
29
21
17
12
5
9
6
6
4
3
1

-*

3

N

•^"
O

in

O

in

O

in

O

tC

o

to

o

to

o

to

O
*>

to

C-

O

c-

0

t-
O

00

O

00

o

00

O

o.
O

o»

O

OJ

O

OJ

o

in

00

TH

••*

c—

o

CO

O3

N

1/5

00

1— 1

•**

t—

0

CO

to

•*

•**

•*

in

in

us

tO

tO

to

to

t-

c-

t-

OO

00

OO

Oi

Ol

o>

LEG I.
12.1 to 13.1.. .

1

'3
1

13.1 to 14.1....
14.1 to 15.1

2

1

2
6
3

1

i

1

-6
9
6
2

'i

1

15.1 to 16.1

16.1 to 17.1....
17.1 to 18.1....
18.1 to 19.1....
19.1 to 20.1....
20.1 to 21.1....
21.1 to 22 1. .

••

6
11
13

8

1

3
11
16
7
3
1

2

4
10
20
14
2
3
3

'3

2
3
9
8
6
1
1

i

1

2
8
4
4

1

••

22 . 1 to 23 1

1

1
1

•>

2
1
3

23.1 to 24.1

1

1

2
3
2

5

1

1

24.1 to 25.1....

1

25.1 to 26.1....

26 1 to 27 1

5
1

'3

27.1 to 28.1 . .

28.1 to 29.1

1

29.1 to 30.1

1

30 1 to 31 1

3

31.1 to 32.1

1

32.1 to 33 1

33.1 to 34.1. . .

1

1

Total. ..

2

5

14

25

40

42

58

33

20

10

14

12

5

1

1

1

283

LEG II.

4.9 to 5.2..
5.2to 5.5....
5. 5 to 5.8....

2

2
3

1

1

4
4
3
1

1

6
11

14
24
44
46
41
30

1
3

6

8
4
2

1

4
8
17
9

5. 8 to 6.1....
6.1 to 6.4....
6.4to 6.7....
6.7to 7.O....
7 0 to 73.

4
15
16
5
2

'i

17
25
4
4
1

••

••

1

1
10
14

7
1

1

1

8
7
4

7.3to 7.6....
7 6 to 79

1

1

3
2

4
1

1

22
12
12
11
5
2
1

7 9 to 82

4
3
5

2

8 2 to 85

5
3

2
1

8 5 to 88.

2
1
1
1

8 8 to 91.

9 1 to 94

9 4 to 97

9 7 to 10 0

10 0 to 10 3

1

1

Total .

2

5

14

25

40

42

58

33

20

10

14

12

5

1

1

1

283

LEG III.
6 6 to 70

1
3
1

1

12
16

28
38
48
45
35

7.0to 7.4....

7 4 to 78

2

5
7
1
1

2
5
12

5

3
12
20
4
1

7. 8 to 8.2....
8. 2 to 8.6....
8 6 to MO

2
7
23
8
2

'3
19
24
8
3
1

'i

2
8
16
5
1

i

1

9 0 to 94

2
9

1

1

9 4 to 98

9.8 to 10.2..
10 2 to 10 6

••

1

4

4

3
3

3

1

17
13
12
9
6
9

4
5
5

10 6 to 11 0

4
3
4

ll.Otoll.4...
11 4 to 11 8

••

1
2
o

11 8 to 12 2

12 2 to 12 6

12. 6 to 13.0....

1

1

Total

2

5

14

25

40

42

58

33

20

10

14

12

5

1

1

I

1

283

68

VARIATION AND CORRELATION IN THE CRAYFISH.

TABLE 31.

LENGTH OF
PROPODITE.

LENGTH OF PROPODITE, LEG I.

"3

«

•**<

lO

CO

t~

CO

Ol

a

0

T— 1
M

O

cvi

CM

O

co
CM
0

-*

CM

0

8

3

8

o

r—

C>)

O

CO
CM

O

8

0

o

CO
0

*— i

CO

O

co1

0

s

0

3

0

0

n

o

0

o

0

o

N

CO

-t

in

co

t—

od

OJ

S

rH
CM

CV>
N

CO
CM

-*r»

CM

£

CO
C^l

£1

CO

evi

OJ
CM

S3

53

%

S?

1

LEG II.

4 9 to 5 2

1

1
6
11
14
24
44
46
41
30
92
12
12
11
5
2
1

5 2 to 5 5

2
1

2
4

2
5
4
2

5 5 to 58

1

5 8 to 61

9
9
1

1

6 1 to 64

8
11

5

4
16
9
3

1

6 4 to 67

12
18
4
2

4
11
13
2

6 7 to 70

g

19
8

'2

15
4

1

7 0 to 73

7 3 to 76

2
10
3
1

1

6

4

7 6 to 79

9

7 9 to 82

2
2

1

2
6

5'
1

1
1

3

1

1
1

• •

• •

• •

8 2 to 85

8 5 to 88

1

8 8 to 91

1

2

l
l
1

1

• •

9 1 to 9 4

9 4 to 97

9 7 to 10 0

10 0 to 10 3

1

1

Total

1

4

6

13

19

24

34

37

31

29

21

17

12

5

9

6

6

4

3

1

••

1

283

LEG III.

6 6 to 70

1

1

12
16
28
38
48
45
35
17
13
12
9
6
2

7 0 to 74

1

3
1

1

4

4
6
3

2
4
9
3
1

1

7 4 to 78

1

7 8 to 82

10
8
3
2

5
13
11
3
2

1
12
15
6
1
1

8 2 to 86

1

13
15
1
1

1

8 6 to 90

4
15
10

1

9 0 to 94

3
15
2

1
4
8
2

T

9 4 to 98

2
5
4

9 8 to 10 2

10 2 to 10 6

4

1
4
3
1

1
4
1

3
3

1

Y

2

Y

2

Y

••

• •

10 6 to 11 0

11 0 to 11 4

1

11 4 to 11 8

1

11 8 to 12 2

12 2 to 12 6

12 6 to 13 0

1

1

Total

1

4

13

19

6

24

34

37

31

29

21

17

12

5

9

6

6

4

3

i

1

283

TABLE 32.

LENGTH OF
PROPODITE.

LENGTH OF PROPODITE, LEG II.

IM

in

00

1-H

-*

C-

0

CO

CO

oa

e>i

m

CO

t-H

•*

t-

0

CO

"3

•u

I

in

S

05

in

o

4J

CM

in

0

+3

m

to
o

*-

CO

«o

o

-t»

T— 1

C0

5
•>*

t—

o

-tJ
t*

c—

a

o

t-

o

-u

CO

t-

O
-w

CO

00

o

43
O>

00

O
-M

CO

CO

5

o

oi

3

CO

o>

_0

rl

a.

5
-*

o

rH

O
-t>
l~

o

o

49

O

**

in

m

m

•J3

CO

CO

t—

t—

t~

c-

CO

00

CO

O)

0>

05

o

rH

LEG III.
6 6 to 70

1
3
2

l
12
16
28
3*
48
45
35
17
13
12
9
6
2

7 0 to 74

1

2

8

1

4
4

5

1

2
1
15
3
2
1

7 4 to 78

'7

22
14
1

1

7 8 to 82

8 2 to 86

9
25
11

i

6
26

7
1

1
1

5

20
3

1

8 6 to 90

9 0 to 94

-8
12
1

1

9 4 to 98

9 8 to 10 2

1

9

10 2 to 10 6

3

7
2

10 6 to 11 0

4
5
2

1

1

3

Y

i

Y

••

11 0 to 11 4

1

11 4 to 11 8

11.8 to 12 2

12 2 to 12 6

12 6 to 13 0

1

1

Total

1

6

11

14

24

44

46

41

30

22

12

12

11

5

o

i

1

283

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Mill. WHOI LIBRARY

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