142 



PROBLEMS OF RELATIVE GROWTH 



time which elapses between the origin of the body and that 

 of the organ. Then the length of this period is, by Schmal- 

 hausen's method, measured by the linear increase of the body : 

 let this be denoted by L x . Then if the weight of the organ 

 at times t and t x be v and v lf the linear dimensions of the 

 body, not for the same time, but for the corresponding phase 

 of its development, will be (L — Lj.) and (L x — L x ) ; and the 

 corresponding body-weights (since the linear dimension is 

 derived directly by taking the cube root of the weight) will be 

 as the cubes of these values. 



Thus the growth-quotient q for the organ for this period 

 of time can be written 



a= log Pi -log* _ ( () 



* 3 [log (U - U) ~ log (L - L x )] w 



The corresponding formula for linear measurements of the 

 organ will of course be the same, but with the omission of 

 the 3 in the denominator. 1 



As example Schmalhausen takes the length of the parts 

 of the hind-limb of the developing fowl. Here the develop- 

 ment begins proximally, so that e.g. the most distal (4th) 

 phalanx of the 3rd digit begins to develop two days later than 

 the 1st or most basal. 



Taking simply the initial and final values for length between 

 the 8th (or 9th) and the 21st day, he arrives at the following 

 result. 



Femur 



{Phalanx 1 

 3 

 4 



Origin at linear 



size (L) of embryo 



mm. 



377 

 5-20 



6-87 



8-25 



8-89 



k calculated from 



Huxley's hetero- 



gony formula 



q calculated from 

 the linear modifi- 

 cation of formula 

 (4) above 



1-43 

 i-6 5 



1-75 

 1-64 

 1-85 



i-li 



i-oo 

 0-84 

 o-86 



1 If we knew the time-relations precisely, the proper formulation of 

 the growth quotient would be 



= log v t - log v 



^ " " log (w 1 — w x ) — log (w — w x ) 



where w x is the amount of weight added by the body in the period t x , 

 between the time of its origin and that of the origin of the organ. 

 Mathematically, it may be pointed out, this is not identical with 

 expression (4) ; but the latter gives a reasonable approximation. 



