4 
were measured from the front edge of the eye in order to be able 
to include published measurements. The eye is a well-defined 
base of reference for measurements in fishes and is particularly 
useful in growth studies since it is differentiated very early 
and is firmly anchored in the chondrocranium, 
Differential growth.—In the accompanying figure, p. 5, 
the measurements of the Length of rostrum are plotted against 
those of the length of body using the same logarithmic scale. An 
examination of the distribution of these points suggests a curve 
rather than a combination of two or three straight lines. Hux- 
ley (1932) has described a number of relative growth curves in 
which one straight line breaks abruptly into: another straight 
line with a different slope. He also shows instances in which 
one straight line breaks into another through a short curve. May 
it not be that all such relative growth curves are hyperbolas 
with asymptotes crossing at varying angles and approaching these 
asymptotes more or less closely? It appears from inspection that 
the Folyodon measurements lie on an hyperbola whose asymptotes 
cross at an angle of about 135 degrees. A smoothed curve has 
been drawn by selecting the point where curvature is greatest 
and by drawing a symmetrical curve to the rather straight limbs 
at either end. The upper end of the curve, representing measure- 
ments of the larger specimens, is almost straight with a slope 
of 0.68. The lower end has a slope of about 4.1. 
In order to learn something about the distribution of 
this differential growth within the rostrum of Folyodon, the length 
from the barbel to the eye was plotted against body length. This 
region was found to grow at the same rate as the body. Therefore, 
from the standpoint of relative growth, this part between the 
barbel and the eye will be considered as part of the body, since 
all of the heterogonic growth occurs in the part distal to the 
barbels. 
The distance from the posterior edge of the eye to the 
tip of the opercular flap was measured on the specimens in the 
collections of ‘the Illinois Natural History Survey and combined 
with other measurements made in the same way by Danforth (1911). 
When the logarithm of this length is plotted against the logarithm 
of body length one obtains a curve of heterogonic growth similar 
to that for the rostrum excerpt that it shows smaller departures 
from isogonic growth. The slope of the lower limb of the operculum 
curve is about 1.9. It reaches a value of 1 at about the same body 
size as does the rostrum curve, and then declines slightly. 
Barbel length was also measured throughout a wide range 
of sizes of spoonbill. When the logarithm of the length of barbel 
is plotted against the logarithm of length of body it may be seen 
that its relative rate of growth is at all times slower than that 
of the body within the size range of fishes used. The barbel 
curve begins with a slope of 0.7 but, in body lengths exceeding 
100 millimeters, it has a slope of only 0.3. It should be mention- 
ed that the barbel, like the rostrum.and the opercular flap, is 
