638 



THE AMERICAN NATUEALIST [Vol. LI 



any simple, readily integrable theoretical curves. We shall hence 

 be compelled in this case to make use of the expedient so fre- 

 quently employed in applied mathematical problems of all sorts, 

 namely, to take finite summation as a sufficiently close approx- 

 imation to integration. Doing so, we may take as the simple ex- 

 pression of total inbreeding, to and including the nth generation, 

 the following: 



100 Fr,/ 



(i) 



where 2 denotes summation of all values between the inclusive 

 limits indicated, and Ft„ is a constant having the value set forth 

 in Table I. Ft,,, is of course the total area of the maximum 

 brother X sister curve up to and including the n + 1-th genera- 



Sin 



tabled ( 



for all. 



constant they may be 



3 2 125 



4 3 212.5 



5 4 306.25 



6 5 403.125 



7 6 501.5625 



8 7 600.78125 



9 8 700.390625 



10 9 800.19531254 



11 10 900.09765625 



12 11 1000.048828125 



13 12 1100.0244140625 



14 13 1200.01220703125 



15 14 1300.006103515625 



16 15 1400.0030517578125 



In using the form of total inbreeding coefficients shown in (i) 

 there is one caution which must be carefully observed. This is 

 that only so many generations should be used as to include the 

 one in which the observed Z taken first reaches its highest value 

 for the pedigree under discussion, and not any beyond that one. 

 This will usually be for the earliest ancestral generation of the 

 pedigree, but not always. 



