254 



Journal of Agricultural Research 



\'ol. Ill, No. 3 



over the range covered. The values at the lower temperatures obtained 

 in this way will, of course, not be as near the truth as would be the case 

 if densitv determinations were made at the low temperatures; but, on 

 the other hand, to include determinations at the lower temperatures 

 would render less accurate the reduced values at the higher tempera- 

 tures — that is, the assumed equation would not come so near expressing 

 the actual rate of expansion over the temperature range where accuracy 

 is most desired. For that reason in making the "least squares" reduc- 

 tion of the obser\'ations at the various temperatures, only the observed 

 densities at 20°, 30°, 40°. and 50° C. were considered. 



CALCULATION OF RESULTS 



After completing the observations g^ven in Table I at as many points 

 as desired, the density at each point was calculated, and from the density 

 values at the different temperatures the rate of change was determined. 



For convenience in calculation it was assumed that the rate of change 

 of density with change of temperature could be expressed with sufficient 

 exactness by means of an equation of the form 



Dt = D^ + a{t-x)+^{l-x)^ + r{t-xy + 



in which £>, = the density at any temperature /, 



Dx = the density at some standard temperature x, 

 n-, ,i, and ;- = constants to be determined for each sample investigated. 



In practice it was found that for certain temperature ranges the 

 expansion was represented within the limits of experimental error, by 

 the above equation, with all terms above the second power omitted. 

 By means of a "least squares" method the observations on each sample 

 given in Table I have been reduced and the calculated values of Dx, «, 

 and 3 are given in Table IV. Observ'ations and calculations of density 

 for an average sample of cream are given in Table III. It will be seen 

 from the closeness of the agreement between the calculated and the 

 observed values of the density at various temperatures that the assumed 

 equation comes very near expressing the actual rate of expansion of the 

 diJBferent samples at the time the density determinations were made. 



T.\BLE I. — Observed densities of milk and cream 



1913- 



July 12 . 



July II. 



Temper- 

 ature. 



"C. 



20 



30 

 40 



5° 



10 



20 



30 

 40 



5° 



Density. 



G.lc. c. 

 I. 0381 

 I. 0368 



I- 0356 

 I. 0322 

 I. 0284 

 I. 0236 



I. 0348 



I- 0343 

 I. 0348 

 I. 0317 

 I. 0292 

 I. 0258 

 I. 0212 



1913- 



Junes. 



June: 



Nov. 19. 



Density. 



G.jc. c, 

 I. 0363 

 I. 0348 

 I. 0320 

 I. 0284 



1.0311 

 I. 0290 

 1.0237 



I. 0314 

 I. 0279 

 I. 0238 

 I. 0191 



o The percentages of (at here given are as reported by the Bureau of .A.aimal Industry at the time the 

 sampU^s were preparetJ , 



