104 ANALYSIS OF THE FOUR PRINCIPLES. 
we obtain the principal plus the interest allowed on the money with- 
drawn by taxation; P = the whole value of the endowment at the 
end of a given number of periods; C = the whole of the money with- 
drawn by taxation and subjected to a separate rate of interest, down 
to a given period.* It should be observed that, as in the bionomic 
problem MW and m are liable to become fractions less than 1, so also in 
the problem of money investment either of these factors may fall 
below 1. This is the case when the charges for management, etc., are 
more than the interest. 
Table giving Formulas for the Ratios between Cross-breeds and Pure-breeds. 
In third generation, P = A (M — Mc)s = pure-bred individuals in 
the third generation, 
In nth generation, P = A (M— Mc)* = pure-bred individuals in 
the nth generation. 
C —the number of cross-bred individuals in any generation. 
In first generation: 
€—Acm 
In second generation: 
C= Acm2 + Acm (M— Mc)! 
In third generation: 
C= Acm3 + Acm? (M— Mc)! + Acm (M— Mc)? 
ns fps TES OR [cameras uni Ha Na 
= Acm (M Mc)? \ | M— Mc ii | M— Mc | F I J 
In nth generation: 
( 2 
C= Acm(M— Mc)" \ C)ealee ser a= Me | 
NE TS aoe ea 
J j 
In third generation: 
6 IM En Ree oe a cae es \ 
P~M—Mc™ \ |M—Mc | AS | M— Mc | ne 
* [he method by which the first steps are made in reaching the desired formula 
will be understood by considering this endowment problem, The advantage of 
the formula here reached is that it gives the ratio of all the cross-breeds to pure- 
breeds, and not simply of half-breeds to pure-breeds, as was the case in the formula 
reached in my paper on Divergent Evolution (see Appendix I). 
+ This is obtained by dividing each term of the second member of the previous 
equation by Acm (M — Mc)?, and then placing the same amount as a multiplier 
outside of brackets. 
