ADDITIVE SCALES 



covariances in terms of which we can express the genetical 

 properties of our faniihes. This is most easily done if the various 

 agencies, genes and non-heritable factors, add together in their effects 

 on the phenotype. That is to say, if the difference produced in the 

 phenot^'pe by substituting one allelomorph of a gene for another, 

 or one environment for another, is the same no matter what the 

 effects of the other genes or environmental factors may be. This 

 may, of course, be merely an ideal, for it may be impossible to find 

 a scale on which all genes supplement one another's action in an 

 additive way. Since, however, we cannot isolate the genes and 

 discover their individual properties we can measure only their 

 average behaviour, A scale will, therefore, be satisfactory on which 

 the gene and non-heritable effects are simply additive on the average. 

 Now, if we cross two true-breeding lines and then backcross 

 the Fi to one of them, mendelian theory shows us that the backcross 

 will contain equal numbers of individuals homozygous, like the 

 parent to which the backcross is made, and heterozygous, like 

 the Fi, in regard to each gene by which the parents differed — 

 equal, that is, apart from sampling error. Then the average 

 expression of the character in the backcross, will, in so far as any 

 one of the genes is concerned, be mid-way between the parent and 

 the Fj. If, and only if, the effects of the genes are additive on the 

 scale used, will this relation also hold between the parental rriean, 

 the Fj mean, and the backcross mean, for all the genes taken 

 together. In other words the two backcrosses provide us with two 

 tests of adequacy of the scale. Letting B^ be the mean measurement 

 of the individuals in the backcross to the parent whose mean is Pj, 

 the scale must be such that : — 



2B1 = Pi + Fi and 2B2 = P2 + F^ 



In the same way one quarter of the individuals in an F2 are like 

 each parent and half like the Fj in respect of each gene, so that if the 

 effects of the genes are additive 4F2 ^ 2F1 + Pi + Pg = 2B1 -j- 2B2. 

 Similar tests of the scale can be devised using F3, the mean measure- 

 ment of Fg, and so on. 



The test of the additiveness of heritable and non-heritable con- 

 tributions to the variation is somewhat different. The variation of 



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