VII RELATIVE GROWTH 237 



The sign o{ y" corresponds to the sign of (ace^^'-i) = ^j; (t). ^j; (/) is a monotonically 

 decreasing function passing, between t = o and t = ^, the values from (a-i) to — i. 

 Hence, for a < i , j" is always negative, i.e. the curve has no inflexion for positive values 

 of t and is qualitatively not much different from the curve for j' = 9 (<). For a > i, the 

 curve has an inflexion. Here the second derivative o^ y disappears : 



nrp~l<ty I ^ n 



or, settmg 



^-kt, — 



9 ih) = I 



9(<i) 

 I 



Hence : 



(^1) 



yih) 



Zi(«) 



/ ('1) = \—-J ='^X2 (^ 



The course of the functions ^j {a) , ^2 (<^) > for <^ > i continually variable, is apparent from : 



XlW 



(a-i) 



X2 («) = ,^ « 



with the limiting conditions: 



\vm.i^ (a) =0; 

 a — » I + o 



lim -/2 (a) = I 

 a -^ I + o 



a -^ 



lim '/2 (a) 

 a — > =0 



The curves of j^j, '/2 (Fig. 39) determine the ordinate of the point of inflexion and the 

 slope of the tangent at this point, respectively. The abscissa of the point of inflexion is 

 determined by: 



hence 



^(^1^ 



t, = 



-kt. 



In (ca) 



1 2 3 



Fig. 39. Explanation see text. 



Hence, tj is positive for ca > i, and negative for ca < i. The latter case gives a curve 



which is qualitatively similar to the monotonic asymptotic rise of 9 = i-cer'^' for positive /. 



In a growth curve starting at an early age, x^ will be small compared to .v*, and hence 



c » I . Depending on the value of a, the curve of time growth of an organ will either show 



Literature p. 253 



