74 
GENETICS: W. E. CASTLE 
Proc. N. a. S. 
excess of 50. But further consideration of the matter leads me to with- 
draw this objection. If the arrangement of the genes is really linear, as 
assumed by Morgan, then double crossing-over is a sufficient explanation 
of why cross-over percentages do not increase as fast as map distances. 
But double crossing-over is the questionable phenomenon which need 
not be assumed if the arrangement is non-linear. Hence, by constructing 
models such as are shown in figures 1 and 2, I have attempted to ascertain 
m r 
FIG. 1 
whether a workable system could be found which was able to dispense with 
double cross-overs, assuming a non-linear arrangement. It seemed to me 
that such was the case even after the model seen in figures 1 and 2 had been 
constructed. I next examined the data given by Gowen^ for the linkage 
relations of eight genes of the "third chromosome" of Drosophila. On 
constructing a model to show the spatial relations of the genes, without 
assuming the arrangement to be linear, and without making any allowances 
for double or triple crossing-over, I obtained a figure strikingly similar to 
