94 



SCIENCE 



[N. S. Vol. LIV. No. 1387 



fer by two independent but equally powerful 

 factors, neither of wliicb shows dominance. 

 Fj will again be intermediate but of a single 

 type and not more variable than either pure 

 parent race. But F,, by recombination of 

 the two differential factors, will now consist 

 of five graded types, two of which correspond 

 with the parental types, while the remaining 

 three are found in the intervening region at 

 equally spaced intervals. If the several 

 graded types are readily distinguishable one 

 from another, they will be found to occur in 

 the proportions 1:4:6:4:1. But if the 

 types are so close together in appearance as 

 not readily to be distinguishable, the distribu- 

 tion will resemble a probability curve. 



Further, if three independent but equivalent 

 factors are involved in a cross where domi- 

 nance is wanting, the F^ classes will number 

 seven and their frequencies will be as 

 1 : 6 : 15 : 20 : 15 : 6 : 1. 



Now, suppose that in these several hypothet- 

 ical cases, the character under investigation 

 is size, and that the amount of difference in 

 size between the parents is in every case the 

 same; let us say for convenience, 12 units 

 (inches, pounds, or whatever the case may be). 

 Then the several classes of individuals of 

 which F„ is composed will have the class mag- 

 nitudes and frequencies shown in Table I. 



For distributions, such as those shown in 

 Table I., we can readily calculate standard de- 

 viations, which measure the variability of each 

 array. See the last column of Table I. It 

 will be observed that the standard deviation 

 falls off rapidly as the number of factors in- 

 volved increases. Inspection of the column 

 headed " standard deviation " in Table I. will 

 allow one to arrive at the law of decrease of 

 the standard deviation with corresponding in- 

 crease of factors. It is evident that as the 

 number of factors is doubled, the standard de- 

 viation is halved under the radical sign. In 

 other words, to reduce the standard deviation 

 hy one half, the numher of factors must he 

 increased four fold. With this point in mind 

 one can extend as far as is desired the col- 

 umns in Table I. headed " factors " and " stan- 

 dard deviation." 



In Table I. the difference between the pa- 

 rents is assumed to be 12 units and the stan- 

 dard deviation is expressed in terms of those 

 units. To give the table a general form, we 

 might suppose the difference between the pa- 

 rents to be one unit. The standard deviation 

 would then be only one twelfth as great. It 

 is so given in Table II., wherein only the 

 columns " factors " and " standard deviation " 

 are entered from Table I. 



TABLE n 



Standard Deviation of F^ Expressed in Per Cent, 

 of the Difference between the Parent Baces 



In the foregoing discussion, it has been as- 

 sumed that the parent races were completely 

 homozygous and so deviod of genetic varia- 

 bility. If this were true of the parents, it 

 would also be true of Fj. In that case what- 

 ever variability was exhibited by the parents 

 or Fj would be non-genetic. Under like en- 

 vironment F„ would be expected to show a 

 like amount of non-genetic variability. Hence 

 in estimating the genetic variability of Fj 

 one would have to deduct from the total ob- 

 served variability of F, an amount equal to 

 the observed (non-genetic) variability of F^. 



In practise one would proceed as follows. 

 First find the difference between the standard 

 deviations of F^ and F^. Divide this by the 

 difference between the parental means (the 

 respective means of the two pure parent races). 

 Multiply the quotient by 100. Kow look in 

 Table II. for the nearest corresponding num- 

 ber in the column "standard deviation." Op- 



