268 EXPERIMENT STATION RECORD. [Vol.43 



Is the arrangement of the genes in the chromosome linear? W. E. Castle 

 (Proc. Natl. Acad. ScL, 5 (1919), No. 2, pp. 25-32, figs. 2).— The author suggests 

 that tlie mutual crossover values of three-linked genes may be to each other as 

 the sides of a triangle, and presents a diagram showing that this hypothesis 

 gives concordant results (barring one conspicuous exception) when tested with 

 data collected l)y Morgan and his collaborators as to the linkage relationships of 

 the genes associated with the sex chromosome of Drosophila melanoga.Hter. It is 

 considered difficult to explain this resiilt on the theory of the linear arrangement 

 of genes on a chromosome. 



The spatial relations of genes, A. H. Sturtevant, C. B. Bridges, and T. H. 

 Morgan (Proc. Natl. Acad. Sci., 5 (1919), No. 5, pp. 168-173).— A reply to Castle, 

 in which the authors refuse to admit tl^e possibility of a three-dimensional ar- 

 rangement of a group of linked genes. 



Are genes linear or nonlinear in arrangement? W. E. Castle {Proc. Natl. 

 Acad. Sci., 5 (1919), No. 11, pp. 500-506). — A reiteration of the author's theory 

 of the nonlinear arrangement of genes. 



Are the factors of heredity arranged in a line? H. J. Muller (Amer. Nat., 

 54 (1920), No. 631, pp. 97-121, figs. 4)-— A reply to Castle, in which the author 

 emphasizes the difficulties of combining linkage values derived from different 

 breeding experiments. 



The measurement of linkage, W. E. Castle (Amcr. Nat., 54 (1920), No. 632, 

 pp. 264-267). — The author suggests that linkage be measured directly on a scale 

 in which linkage strength is 100 when no crossovers occur and zero when there 

 Is 50 per cent crossing over. " Linkage strength " is defined as double the 

 difference between 50 and the percentage of crossovers. 



The probable errors of calculated linkage values, and the most accurate 

 method of determining gametic from certain zygotic series, J. B. S. Hal- 

 DANE (Jour. Genetics, 8 (1919), No. 4, pp. 291-297). — Considering two linked 

 hereditary factors, the author derives formulas for the probable errors of their 

 crossover values and reduplication values, (1) when these values are determined 

 directly from a back cross of Fi on the double recessive, and (2) when they are 

 calculated from an F2 array by a special method considered by the author to 

 give the most probable values. The values based on the F2 are found to be nearly 

 as accurate as those derived from the back cross except in cases of strong 

 repulsion. 



The combination of linkage values, and the calculation of distances be- 

 tween the loci of linked factors, J. B. S. Haldane (Jour. Genetics, 8 (1919), 

 No. 4, pp. 299-309, fig. 1). — The author presents a general formula to show the 

 mutual relationships between the crossover values of three linked factors, and 

 derives a differential equation whose solution expresses the distance between 

 the loci of two factors on a chromosome as a function of the proportion of 

 crossing over between them. The differential equation is solved only for the 

 limiting values, the solutions being x=y and x= — i loge (l—2y), where x is the 

 distance and y the crossover proportion. 



The first solution, which is Sturtevant's original hypothesis, would be ac- 

 curate if the chromosomes were rigid rods, but it is in any case a close approxi- 

 mation whenever the proportion of crossovers is small. The second solution 

 implies complete flexibility in the chromosome (i. e., secondary, tertiary, etc., 

 crossovers occur strictly according to the laws of chance), and is approxi- 

 mated in practice when the distance is large. The author shows that the 

 second solution flows from Trow's form of the reduplication theory (E. S. R., 

 28, p. 876). For the sex chromosome of Drosophilla melanogaster, a weighted 



