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W. C. VISSER 



Growth tactor X 



Growth factor X 



Fig. I. Shape of the growth curves according to the law of diniimshing returns, or 

 Mitscherhch equation, for two growth factors x and y. Exponential approach to 

 asymptote. All curves have same shape if expressed in percentage of maximum yield. 

 Fig. 2. Second solution of the differential equations of the Mitscherhch assumption. 

 Yield equation for substituting, or antagonistic, factors. All curves have same shape 

 but are shifted to the left. 



In formula 1.4 the logarithm of the yield is an additive function of the 

 influence of any separate growth factor. In formula 1.5 the same is true for 



the yield deficit. 



The curves in Fig. i depicting formula 1.4 may be derived from each 

 other by vertical change of scale. The curves in Fig. 2, depicting formula 

 1.5, may be derived by horizontal translation of any of these curves. 



The Projective Yield Function 

 The projective function is often considered as an empirical formula. It may 

 be proved, however, that the formula is based on the assumption that the 

 yield increase is directly related to the yield deficit and inversely related to 

 the level of fertiHty. 



The following equations result : 



differential equation growth function 





x+B 



(2.1) 



q^A 



x—c 

 x+B 



(2.2) 



typical form 

 A X . B 



A-a B-c B-c 



(2.3) 



Multifactorial equation 

 ist solution 



= A ^ 



y-c^z—c^ 



x+Bj^ Y+B2 2-+B3 



(2.4) 



