L02 



THE AM KMC AX XA T U BALIS T 



[Vol. LIV 



distances and the per cents of separations. Just as dis- 

 tance AB plus distance BC on a line are equal to distance 

 AC, so the corresponding small frequency of separation 

 between A and B, plus the small frequency between B 

 and C, are found to be almost exactly equal to the fre- 

 quency of separation between A and C; for this reason 

 if the factors A, B and C are represented as points in a 

 straight linear map, the distances between any two of 

 them will represent the corresponding separation fre- 

 quencies in an almost proportionate manner. A few ex- 

 amples of this principle are shown in Table I ; it has been 

 confirmed in innumerable other crosses, with many dif- 

 ferent factors. Moreover, it is found that the smaller 

 the distance involved, the more exact is the porportion- 

 ality that obtains, the less being the relative discrepancy 

 between the frequency AC as found by experiment, and 

 the value AC obtained, as on the map, by the summation 

 of values AB and BC. The relationship which exists be- 

 tween the small separation values is hence just the sort 

 which Castle himself would demand, for a proof of linear 

 linkage. But whereas Castle would require this relation- 

 ship to. hold for all values, small or large, it may be 

 show that its existence in the case of the small alone is 

 all that would be necessary for a complete proof of the 

 doctrine of linear linkage, even if the large values were 

 no sort of function of the linear series. For, if we pro- 

 ceed according to Castle's own method, and construct a 

 map to represent the relations of the small values just 

 described, showing each of the frequencies by a propor- 

 tionate distance on the map, we necessarily obtain a map 

 each section of which is practically a straight line. In 

 the case, for example, of the data for v, g, and f , shown in 

 Table I, if we represent the separation frequencies by 

 proportionate distances in space, we must place point v 

 at 10.7 units from g, and g at 11.3 units from f ; if these 

 two conditions are both to hold, then the only possible 

 way of bringing f to its distance of 21.8 units from v is 

 to put the three points in a nearly straight line, as shown 



