Vol. 6, 1920 
GENETICS: W. E. CASTLE 
73 
Mr. W. E. Kdington, graduate student in the University of Illinois, di- 
rected my attention to the desirability of a study of the given general 
equations as regards commutative operators with a view to simplifying 
the treatment of the generalized polyhedron groups. 
MODEL OF THE LINKAGE SYSTEM OF ELEVEN SECOND 
CHROMOSOME GENES OF DROSOPHILA 
By W. E. CASTI.K 
BussEY Institution, Harvard University 
Communicated December 26, 1919 
In previous papers^ reasons have been given for thinking the spatial 
relations of genes to be three-dimensional rather than linear as assumed 
in the hypothesis of Morgan and his colleagues. In particular it has been 
shown that in models made to represent the spatial relations of the genes 
in the sex chromosomes of two different species of Drosophila, using the 
original unmodified data, as reported by Morgan and Bridges^ and by Metz,^ 
the arrangement is apparently tliree-dimensional. Recently data have 
become available for modeling in a similar way the interrelations of the 
genes in the so-called "second chromosome" of Drosophila ampelophila.* 
Two views of such a model are seen in figures 1 and 2. The model 
shows the linkage relations of eleven genes, those for which the data are 
most complete and reliable according to Bridges and Morgan. The data 
are taken from table 140 of the publication mentioned. Bach gene is 
represented in the model by a small ring of wire and it is connected with 
another gene by a wire as long in inches as is the cross-over value, in per 
cent, between the two genes joined by the wire. In the figures the model 
is suspended by S, the view shown in figure 2 being taken at right angles 
to that shown in figure 1. The eleven genes represented in the model 
are as follows : 
S, star (0.0) vg, vestigial (65.0) 
Sk, streak (15.4) c, curved (73.5) 
d, dachs (29.0) px, plexus (96.2) 
b, black (46.5) a, arc (98.4) 
pr, purple (52.7) sp, speck (105.1) 
mr, morula (106.3) 
The numbers in parentheses indicate the position of each gene in the 
linear "map" of Bridges and Morgan (p. 127). It will be observed 
that seven of the genes in this list are represented in the "map" at 
distances greater than 50 from star, which lies at 0. In the previous pub- 
lications already cited, I have maintained that since cross-over per- 
centages of 50 or more have not in any case been observed and are logically 
impossible as a result of linkage, map distances should not be shown in 
