Substituting the values of /u and c previously determined: 



1.56 

 (VI) *S^j_ = .279 = 5.9? billions 



U.98 

 (VI) S.5 = ■."SW » 7.30 billions 



These stocks may now be broken dovm into their component ages in 

 accordance viith the previously determined relative age compositions. 



Table 3. — Numbers of fish of each age obtained 



Since the fish of age one are not fully available to the fishery, 

 it is necessary, in order that the recruitment to age one may be deter- 

 mined, to work-backward from the first age class that is fully available, 

 age three In our example. Let s^ and s^ be the available stocks of any 

 two successive age classes, and 5]_ and S2 be the total stocks of these 

 two age classes, respectively. If the stock of the second age class is 

 considered fully available, Sg = S2. s-j^ is subject to the full mortality 

 rate a, but the balance of the age class (S]_ - s-[_) is subject only to 

 natural mortality n. Formulating these statements we have: 



Sg = S2 = S]_ - as]_ - n(S-, - s-, ) 



Expanding: S2 =^ S-|_ - as-, - nS-, + ns. 



Transposing; S2 + as-|_ - ns-j_ = S, - nS, 



Factoring:. 



So -♦■ 



s^(a - n) = S-i_(l - n) 



so + S-. (a - n) 

 Dividing by (1 - n): S-,_ = -^ — T~-li 



(VII) 



Substituting the empirical values previously determined and letting 

 subscripts refer to ages, we find for the first period: 



ions 



r^TTT\ Q 2.0k -f .68('.liO - .13) 



(VII) S2 = 1 - .ij = 2.55 billi 



(VII) Si = 2,55 \^j;Q|£^Q - -^^^ = 2.99 billions 



176 



