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 A/ and m are liable to become fractions less than i , so also in 

 the problem of money investment either of these factors may fall 

 below i. 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 Me) 3 = pure-bred individuals in 

 the third generation. 



In nth generation, P A (M Mc) n = pure-bred individuals in 

 the nth generation. 



C -- the number of cross-bred individuals in any generation. 

 In first generation: 



C=Acm 

 In second generation: 



C = A cm 2 + Acm (M Me) ' 

 In third generation: 



C = Acm 3 -f Acm 2 (M - Me) 1 + Acm (M - Me) 2 



= Ac* (M - Me)' { \ ff^] V f ,-^Vj ' + , } t 



I i .vl -\ 1 t I i . V7 . V/ tr i I 



In nth generation : 



C=Acm (Af-Mc)*-' { ()-' - - ()*+ L^- J *+ 



In third generation : 



m 



[M-Mcj 



C mc / r m ~} 2 f m 1 , 



P~M-MC X - - 



* The 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). 



t This is obtained by dividing each term of the second member of the previous 

 equation by Acm (M Me)-, and then placing the same amount as a multiplier 

 outside of brackets. 



