204 



THE AMERICAN NATURALIST 



[Vol.L 



question which will be reserved until later. But it is 

 clear that, since coincidence does not vary independently 

 of " distance," a linear relation holds between the linkage 

 values, in that these values can be calculated from a 

 linear diagram much more exactly than would be ex- 

 pected on chance relationships; in mathematical ter-ms, 

 the frequencies of separation between all combinations of 

 the different factors in a group are largely a function of 

 the distance apart of these factors in a linear figure. 



It will now be desirable to consider these same facts 

 from another angle. As it is possible to represent the 

 linkages between any three factors of a group in terms of 

 their distances on a linear diagram, it follows that all the 

 factors of a group together can be represented in one 

 linear diagram. Suppose that such a diagram has been 

 made, and that the order of the factors in it is ABCDEFG. 

 Now, as has just been explained, since per cent. AC nearly 

 equals per cents. AB plus BC, it must follow that a separa- 

 tion between A and B rarely coincides with one between B 

 and C ; the same fact may also be expressed by saying that 

 when A and B separate, C stays with B rather than with 

 A. Similar relations, of course, hold for the other fac- 

 tors, too ; thus D also stays with B and C when A and B 

 separate, but it stays with C when B and C separate. The 

 linkage of D with B, then, is only due to its linkage with 

 C, for, although it usually stays with B, it very rarely 

 stays with it except when C does. Thus D is linked to B 

 only through C, and to A only through B. Similarly, all the 

 other factors also are linked together in a chain, each to 

 the one on either side : just as D is linked on the one hand 

 to E, and on the other hand to C, but not to any other fac- 

 tors except through one of these, so C is linked on the one 

 hand to D, on the other hand to B, but is linked to E only 

 through I), and to A only through B, etc. ; moreover, all 

 the factors are linked to the others in the same order. 

 Separation of factors in such a group accordingly means 

 the breaking of the chain at just one or two points, for it 

 has been pointed out that when B and C separate, A and 



