64 ILUyOIS BIOLOGICAL MONOGRAPHS [306 



clearly borne in mind that these conclusions are based on a comparison of 

 angles presented by different curves in the same horizontal unit. Such a 

 comparison can be directly interpreted without danger of going astray, 

 but in comparing the slopes of parts of the same curve or of portions of 

 different curves in different units, we are confronted with a situation which 

 is liable to be misleading without an understanding of the relation which 

 these units bear to one another. 



For the purpose of analyzing this relation let us suppose that a certain 

 species has a larval life of sixty days, which we will divide without regard 

 to stadia into six equal periods of ten days each. Suppose further that we 

 represent the curve of this species as we have those in Plate I, using for 

 units, however, these six equal periods instead of stadia. Now in this 

 graphic representation, the periods in phylogeny to which these ten-day 

 divisions correspond are given equal value, whereas in reality this is very 

 far from true. According to the general conception of the working of the 

 law of recapitulation, the first ten days would represent a much longer 

 phylogenetic period than the second ten, which, in turn, would correspond 

 to a portion of the race-history of much greater duration than would the 

 third ten days, and so on until the last ten day division, whose correspond- 

 ing phylogenetic period would be, perhaps, but a minute fraction of that of 

 the first ten days. The fact that the change in the position of the setae of 

 the trunk, a recapitulative one, is very much greater from first to second 

 instars, than thruout the entire remainder of the larval life illustrates this 

 principle. To represent graphically this condition it would be necessary to 

 extend greatly the length of the first unit, lengthening the second one to a 

 lesser extent, the third a still lesser amount, and so on. We have no means 

 of knowing what the relative lengths of these units should be in order to 

 render the slopes of different parts of this hypothetical curve exactly repre- 

 sentative of the relative rates at which these changes in epicranial index 

 have evolved during different phylogenetic periods. We merely know in 

 which direction to apply this sort of correction. 



Another means of correction may be applied to this hypothetical curve, 

 by leaving the units equal, as they are in Plate I, but dividing the sixty 

 day larval Ufe into six unequal periods, which gradually increase in length 

 from younger to older. The same result would be accompUshed in this 

 manner as by keeping the periods equal and altering the length of the 

 units, in the manner just described. For mechanical reasons it has been 

 necessary to use stadia for our units in Plate I. As already demonstrated 

 the lengths of stadia generally do not present a gradual increase from 

 younger to older in this family, but are often about equal, except for the 

 last, which is usually much longer, and for the first, which is long in some 

 species. The employment of stadia as units, then, offers no correction, 

 except possibly for the last unit, where this stadium is long. It is question- 



