PROTEIN REQUIREMENTS OF CATTLE: MITCHELL 19 



The fit of the Brody curve to the data of Group III is excellent except 

 for the birth weight (9.4 months) and for the last few months of 

 observation. Again a depression of observed growth of no apparent 

 biological significance is smoothed out, in this case between the 37th 

 and the 51st month from conception. 



The rate of weight change ivith age. — At this point, one of the great 

 advantages of reducing a set of irregular age-weight observations to a 

 simple mathematical expression may be demonstrated. From the original 

 data on the growth in weight of these cattle it would be impossible to 

 compute in a satisfactory manner the rate of gain at any time. The most 

 direct method of making such a computation would be limited in time to 

 the particular intervals of weighing adopted in the investigation, and 

 would yield a set of successive values so irregular as to possess little 

 significance, and hence little value. However, the growth equations 

 derived from each of the three sets of data may be changed by a simple 

 mathematical operation (differentiation) to other equations from which 

 the rate of growth (gain per month in this case) may be computed for 

 any age, and successive values obtained will vary in an orderly fashion, 

 evidently representative of highly significant biological changes. 



Thus, differentiation of Brody's growth curve represented by Equa- 

 tion (1) gives 



dW 



-dT ='^-^^-" (^' 



dW 

 in which — r— is the rate of change of body weight with time at any age t. 



The differential equations for the tliree groups of Missouri cattle are 

 readily made up from Equations (2), (3), and (4) : 



dW 

 Group I "Y^ =122.14e--''3*^* (6) 



Group II -^ = 62.33e--°^°^^ (7) 



dW 

 Group III -^ = 33.62e-°227t (g) 



From these differential equations, the average rate of growth of the 

 three groups of cattle has been computed at intervals of three months, 

 with the results given in Table 6,^ and at intervals of 100 pounds in 

 weight, with the results given in Table 7.^ 



* The first two vahies for Group I have been computed by another method, since 

 the growth curve of Brody did not fit the data well at these ages. It was found 

 that the weight-age data for this group from shortly after, to 9 months after, 

 birth could be accurately described by the parabolic equation W = — 13.42t 

 -f 2.48t^, W being the weight in pounds and t the time in months from conception. 



