100 



THE AMERICAN NATURALIST [Vol. LIV 



may also be expressed as follows : given any three linked 

 factors, A, B and C, if any two of the linkages between 

 them are known— say, the linkage AB and BC— then the 

 third linkage— AC— is determined (the most convenient 

 practical method for calculating it is to make use of the 

 "curve of coincidence" of the particular chromosome). 

 This is, two of the linkage values may be taken as "inde- 

 pendent variables" and the third is then "dependent" 

 on them— in this sense we may say that B is linked di- 

 rectly to A and to C, but that A is only linked to C 

 through the linkage of each of these factors with B. 

 Since this is true of any combination of three-linked 

 factors (ABC, BCD, CDE, ACD, etc.) it can be shown 

 that the factors are all linked together in chain arrange- 

 ment, any one factor being linked directly to only two 

 others (those which we may regard as being on either 

 side of it), its linkage with the rest being entirely de- 

 pendent on these intermediary linkages. 1 This remains 

 true as a discovered mathematical fact of the linkage 

 relationships, shown first in experiments of Sturtevant's 

 designed to investigate the problem, and this is what the 

 writer has designated as "the law of linear linkage." 

 Whether or not we regard the factors as lying in an 

 actual material thread, it must on the basis of these find- 

 ings be admitted that the forces holding them linked 

 together— be they physical, "dynamic" or transcend- 

 ental—are of such a nature that each factor is directly 



iJ. e., all the linkages (factorial (n — 1) in number) between the n 

 factors in a group, can be shown to be dependent on (functions of) only 

 n — 1 ''primary'' or "independent" linkages. To obtain the most perfect 



(what we should call linkages AB, BC, CD, DE, etc., as contrasted with 

 AC, AD, AE, BD, etc.). On this system, the other linkages all become 

 ^'finitely determined, the secondary linkages Wing in each case a function 

 involving the sum of certain of the primary linkages. If, however, the pri- 

 mary linkages are not chosen according to the above rule, so as to consti- 

 tute a "chain formation, no formula can definitely express the relation- 

 ships of the linkages, for the secondary linkages will then in some cases 

 depend upon the sum, in other cases upon the difference between the link- 

 ages taken as primary. 



