Vol.. 7, 1921 
GENETICS: C. B. BRIDGES 
127 
a major fraction of an artificial quota, scaled up or down from the true quota to meet 
the requirements of the computation.) 
The d'Hondt method, originated in Belgium and now widely used in European elec- 
tions, employs "multipliers" 1, 1/2, 1/3, 1/4. . . and can be shown to favor the larger 
states to the extent of violating all four of the conditions expressed in our Fundamental 
Principle. 
2 If one should try to minimize the sum of the squares (or the sum of the absolute values) 
of the deviations of the a's themselves from their true values (with or without "weighting" 
by the population of the state), the resulting methods would all lead to an Alabama 
paradox. The same is true of the weighted sum of the absolute values of the deviations 
oi a/A (or of A /a). The same is also true of the absolute values of the logarithms of 
the ratios between the a'S and their true values. 
3 This Postulate III was added on April 23, after Professor F. W. Owens had shown 
Cat the meeting of the American Mathematical Society on February 26) that the 
method of minimizing the sum of terms like A[(a/A) — ■ (a/A) ] 2 leads to th same 
result as the Willcox method of major fractions. It may be noted that the method of 
minimizing the sum of terms like a[(A/a) — (A /a)] 2 leads, not as one might expect, to 
the method of the harmonic mean, but to the method of the geometric mean. 
CURRENT MAPS OF THE LOCATION OF THE MUTANT GENES 
OF DROSOPHILA MELA NOG A S TER 1 
By Calvin B. Bridges 
Columbia University, N. Y. 
Communicated by T. H. Morgan, December 5, 1920 
The maps that have been published 2 showing the distribution of the 
mutant genes of D. melanogaster can now be much improved because 
of the discovery of new mutants and the accumulation of crossover data. 
Figure 1 gives in simplified form the maps that are in use in our laboratory. 
The distances on the maps are based on the total amount of crossing 
over between the loci, one unit of distance representing one percent of 
crossing over. The map-distances are the same as the observed crossover 
values or "percentages of exchange" whenever the two loci considered 
are so close together that no, or only a negligible amount of, double crossing 
over occurs between them. In the first (X-) chromosome this practical 
equivalence of map-distance and exchange-value holds for loci not farther 
apart than about 15 units. In the middle of the second chromosome and 
of the third chromosome the equivalence holds for only about 10 units. 
In the end-regions of the second and third chromosomes it holds up to 
nearly 20 units. For distances somewhat greater than these the map- 
distances exceed the observed percentages of exchange by an amount equal 
to twice the percentage of double crossing over between the given loci. 
For still greater distances the difference includes also three times the per- 
centage of triples. The number of quadruple crossovers is negligible 
except perhaps when the whole length of the second chromosome is to 
