E. Y, Thomson, J. Bell and K. Pearson 57 
We have accordingly to find Dn-zjDn-i when n is large. Returning to i),i-2. 
subtract /3 times the second row from the first, and /3 times the second column 
from the first. There results: 
7(l+/3^)-2^, 
1-7/3, 
0, 
0, 
.. 0 
7. 
1, 
/3, 
.. /S»-3 
0, 
1, 
7. 
1, 
.. /3»-^ 
0, 
/S, 
1, 
7. 
0, 
l3-\ 
/3-^ 
•• 7 
= {7(1 + - 2/3} - (1 - 7/3)^ D^-.. 
Assume Dn-2 = C^""^ and we have 
^^-{7 (1 + - 2^} 1 + (1-7/3)^ = 0. 
Hence if and ^2 be the roots 
Dn-2 C'lfi'^-^+G^I^^"-^" 
Now if |i be > ^2, this when n is large rapidly becomes equal to 
Thus we conclude that for a great number of generations of inbreeding of 
brother and sister, as will usually be the case in wasps, 
1 
7/3)_j 
where |i is the greater root of the quadratic 
r- {7 (1 + /^^)- 2/3} ^ + (1-7/3)^ = 0. 
The result may be written 
O-nl'Sn = l/Vl+a(fi-7). 
We have worked out this result numerically for a few cases, assuming e the 
coefficient of assortative mating to be = the resemblance of brother and sister 
in most mammals. (This step, of course, from mammals to insects yet needs veri- 
fication. Cf Warren for Aphis and Daphnia, Biometrika, Vol. I. p. 147.) 
Case (i). The average values of biometric work are a = ^, /3 = |. Hence 7 = 1*5, 
1 - 7/3 will be nearly zero and ^1 = -8000. We find 
<x„/s„ = l-23. 
Case (ii). Let the correlations be equal to the gametic correlations on the 
Mendelian hypothesis. Then a = \, lS = ^. Hence 7=1'5 and we have 
-875 1 + •0625 -0, 
which leads to = -7966, and 
o-„/s„=l-24. 
Biometrika vii 8 
