30 
GENETICS: W. E. CASTLE 
must lie laterally about equidistant from both. Other linkage relationships 
of bifid show that it lies as far as possible removed from the plane in which 
most of the genes lie. The other case mentioned is that of abnormal abdo- 
men {A, figs. 1 and 2). Its linkage relations are known with only two other 
genes, yellow and white. But the relation of these two with each other is 
one of the best determined and the linkage of abnormal with each of them 
rests on more than 15,000 observations in each case. The yellow- white 
linkage is 1.1, as already stated; abnormal-yellow is 2.0, and abnormal-white 
1.7. These relations make it impossible for abnormal to lie in line with yellow 
and white. Until a third linkage relation of abnormal is determined, it may 
swing freely round the line which joins yellow with white but can never come 
into line with them. A third linkage relation having once been determined 
for abnormal, its linkage with any other gene in the sex-chromosome could be 
readily predicted from direct measurement of the reconstructed figure. The 
actual test of the utility of this method of portraying linkage relationships 
could easily be made by first forecasting by measurement what undetermined 
linkage values are likely to be and then actually determining them by ex- 
periment. Such predictions could not fail to come nearer the truth than pre- 
dictions based on a linear map, if as I have suggested the arrangement is really 
not linear. 
What, it might be asked, does this reconstruction signify? Does it show 
the actual shape of the chromosome, or at any rate of that part of it in which 
the observed genetic variations lie? Or is it only a symbolical representation 
of molecular forces? These questions we can not at present answer. A first 
step toward answering them will be the construction of a model which will 
give us reliable information as to undetermined genetic relationships. A 
model which will answer questions truthfully must be a truthful presentation 
of actual relationships even though we do not know whether they are spatial 
or dynamic. 
If the arrangement of the genes in the chromosome is not linear, Morgan's 
theory of linkage must be somewhat modified. (1) The fundamental assump- 
tion that the genes lie in the chromosomes and have a definite orderly ar- 
rangement there is not disturbed. (2) The further assumption that the 
respective distances between the genes determine their closeness of linkage 
one with another may also stand unchallenged. (3) But the assumption that 
the arrangement of the genes within the chromosome is linear cannot be ac- 
cepted without proof, which at present is lacking. This assumption has made 
necessary other secondary assumptions, likewise unproved, which are super- 
fluous if this one is abandoned. 
Such an unproved secondary hypothesis is that of double-crossing-over. 
The experimental data show that double-crossing-over must occur, if the 
arrangement of the genes is linear. For if three genes, A, B, C, are linear in 
their arrangement in the order named, and all lie in the same gamete, and if 
subsequently A and C are found together in one gamete and B in another, 
