66 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



Figure 1. — Relation between growth lines. 



estimate of Zi can be obtained as just described, an 

 estimate of L a can be obtained from where the 

 various lines intersect the 45° line drawn from 

 zero, and k can be calculated from L\ and L„. 



The technique described is graphic. The same 

 results can be obtained mathematically. The 

 slopes of the various lines in figure 1 are 



k 3 - 



X.j — lio . ±j 2 — I\ ; I3 — i<2 

 — , Ari== — y — , k 2 = — y — ' 



ii 



J- 1 a I3 i. !•& — L t 



In 



u 



■k n = 



In + 1 J-'n 



In 



Consequently L\- 



_ I>n + \ -^ n 

 K„ 



.(1) 



J £n 1 k n + 1 



But L n =L x , and L n+1 = L X , • 



Subtracting L„ from L n+ i we obtain 



Z 1 *"(l-*)=(l-i)(2c. + i-i 1 



■In + l — ^n 



or Zi = 



But £,= 



k„=k". 



k n 



In + l In -k'n + l I>1 



k„ k" 



•(2) 



; consequently 



