628 RATE OF GROWTH OF DAIRY COW 



— in equation (1) is not expressed as a function of time. Equation 

 dt 



(1) is therefore first integrated. The integration is performed by- 

 writing 



~ = Kx{A - x) 



dt 



in the form 



dx 



dt = 



Kx (A - x) 



which is a standard form ready for integration 



dx 



f.-j 



Kx{A -x: 



Integrating and putting K for KA we get 



1 X 



< = -Iog- +C 



K A — X 



or 



log --^— = Kt + C 

 A — X 



where C is the integration constant. The precise meaning of C, the 

 integration constant, and /, time, may be obtained as follows: Let t, 

 time, be counted from the maximum of the velocity curve (A in the 

 figure) ; that is, where x = A — x. The value of t at the maximum 



point is therefore zero, so is log , and C also, and the equation 



is simplified to A — x 



log -^— = Kt 



A — X 



where i is the time interval between the maximum point of growth 

 and any other time on the curve. Instead of t, we may write (^ — t\) 

 when h is the age of the animal at the maximum point of growth, 

 and t is any age. The equation becomes 



log -^— = K{t- ti) (2) 



A — X 



