The additive theorem o f linkage is only true when the linkage values are small. The number o f recombinations between two genes which lie far apart on the chromosome is less than the sum o f the linkage values for the separate intervals into which the intervening part o f the chromosome can be divided. The explanation o f this can be found in double crossing-over; two simultaneous cross-overs, one in the interval AB and the other in the interval BCy will remove B from between A and C but leave A and C in their original association. The total number o f recombinations between A and C should be compen­ sated for this according to the formula AC = AB + BC — 2AB.BCy since the chance o f a cross-over simultaneously in AB and BC is clearly AB.BCy and each double cross-over must be counted twice.