128 Experimental and Statistical Studies upo7i Lepidoptera 
But before we may examine more closely into the manifestation of selection in 
the present case, certain possibilities of error demand frank recognition. (1) The 
number of individuals is smaller than is desirable for the purposes of statistical 
treatment. It is clear that this error has been allowed for, in so far as each 
comparison has been based upon the numerical values of the types or of the 
variabilities together with their probable errors, and these are determined by 
formulae which take full account of the number of cases under investigation ; and 
besides, the occurrence of selection has been asserted only when the difference 
between comparable indices of vaiiation or between two typical values is in excess 
of thrice the probable error, certainly a generous allowance. (2) In dealing with 
graduated values like the various dimensions of the pupa there must always be 
an error due to the "personal coefficient." Enough individuals have been used to 
render this error negligible, for according to the law of probability the errors in 
excess will be offset by errors in underestimation. (3) A far more important 
objection may be urged against the comparison of the surviving pupae with those 
which were dead when found ; and of course the value of the comparison which 
has been drawn in the present study cannot be considered as final until definite 
data are obtained showing the effect that death exerts upon the pupal characters 
under examination. But the objection can be urged only in certain cases, for the 
rigid nature of some of the pupal structures, like the antenna, is such that little 
or no alteration can follow death* ; and furthermore it does not militate against 
the treatment of pupal-imaginal elimination, for all the individuals there considered 
were alive both at the time of measurement and at the time of metamorphosis. 
(4) Attention has already been dii-ected to the fact that some of the dimensions 
used are only approximations, as the length and proportions of the antenna ; the 
former is really the chord of the arc described by the axis of this organ. But it is 
legitimate to suppose that all chords of a given length represent arcs of the same 
value on the whole, as those arcs with greater curvature will be offset by arcs with 
less curvature, and that therefore a chord of a certain value will correspond to 
a certain average value of the arc. At any rate the differences between com- 
parable determinations of antennal characters in the several groups may be 
regarded as indicating real differences in antennal structure. 
And now, turning to the fundamental question of selection, we must recognize 
clearly at the outset the essential fact that elimination does occur during pupal 
existence and as well at the time of metamorphosis. Those individuals which 
reach the entl of pupal existence form but a portion of the whole number entering 
it, and those animals which reach perfect maturity form again a subdivision of 
the previous group. The fact of reduction in numbers stands unquestioned. 
We have seen that those individuals which successfully survive the severe 
conditions of pupal existence or of metamorphosis are structurally different in 
* I have remeasured dead pupae after a two years' interval and have indeed found certain minor 
differences between the original measurement and the later one. In no case, however, of 70 recently 
remeasured has the difference led to a change of class of the individuals. 
