THE ORDER OF THE GENES 119 



necessary to represent one set of linked genes {a, b, c, etc), 

 ignoring the normal allelomorphic series, for these follow 

 the same (reciprocal) changes. 



If a, h, and c stand for three genes, and if the linkage 

 relations of a to & and of & to c are known, the relation 

 of a to c is a function of the sum of ab and be or of the 

 difference of ab and be. For example, if the crossover 

 value ab is expressed as 5, and that of be as 10, then ac 

 is a function of the sum (15), or the difference (5) oi ab 

 and be. It cannot be said that ae must be 5 or 15 because 

 another possible process may intervene to affect the sum 

 or the difference, viz., double crossing over in the region 

 involved. By making the distance so small that double 

 crossing over is practically excluded the sum or the dif- 

 ference is actually the realized result, as the following 

 example illustrates : 



When three mutant characters yellow, white and 

 bifid were all used together in a single experiment, it 

 was found that there were 1160 non-crossovers, 15 flies 

 representing single crossovers between yellow and white, 

 and 43 flies representing single crossovers between white 

 and bifid. There were no flies representing crossing over 

 in both regions at the same time, i.e., there were no double 

 crossovers. Thus the crossover value yellow white is 

 1.2, and the crossover value white bifid is 3.5. The same 

 data give the yellow bifid crossover value of 4.7, which is 

 precisely the sum of the two component values : 



The simplest way in which such a relation can be 

 thought of is that the three genes stand in a line. Suppose 

 a fourth linked gene, d, is added to the series. It is then 

 found that bd, is a function of the sum or of the differ- 



