COMPARATIVE VARIABILITY IN THE HONEY BEE. 31 



measure many times the number of individuals which were used 

 and to extend the observations over a far greater range of varie- 

 ties. No curve made with 500 individuals would express the 

 true law of variation, nor would ten times that number be suffi- 

 cient, and since the formulation of the law of variation for par- 

 thenogenetic and fertilized forms for this particular kind of par- 

 thenogenesis, arrenotoky, is too important a matter to be based 

 on an inaccurate mathematical formula, it seems better to us to 

 state simply the fact that greater variation does occur in the males 

 and leave the formulation of the law to be worked out with a far 

 greater range of observations and measurements. And then, too, 

 it is by no means certain that this variation follows any fixed law. 

 If the variation were caused entirely by germinal variation, or by 

 any other one factor, then it might be assumed that the law of 

 this variation could be stated in the form of a mathematical for- 

 mula, but as will be shown later, it appears probable that a large 

 part of the greater variability of the drones is due to chance and 

 is therefore not in accordance with any law. It may be argued 

 that variation according to chance is but a way of stating our 

 ignorance of the true law, but if there is a law for this variation it 

 is certainly very obscure, and the working out of this law would 

 require an extremely large number of measurements taken from 

 individuals, each one with its life history known together with a 

 high degree of mathematical ability in its formulation. 



ABNORMAL WINGS. 



As noted previously, in all wings examined, record was made 

 of wings having veins which are not typically found in the bee. 

 Fig. 4 shows, in dotted lines, where these abnormalities occur 

 most frequently. It is very difficult to record these irregularities 

 in any kind of a table, since the irregular veins vary widely in ex- 

 tent and do not arise at exactly the same place in many cases. 

 An attempt was made to classify these according to the veins from 

 which they branch, their extent and direction. Manifestly any 

 tabulation must be considered as merely a matter of convenience 

 in examination. In this we have recorded cases where a vein 

 bends (b in table) from its true course, showing but a tendency 

 toward abnormality, as well as the well-marked cases. The ex- 



