LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN AGRICULTURE M1N CIRCULATING CHECK FOR UNBOUND CIRCULATING COPY UNIVERSITY OF ILLINOIS Agricultural Experiment Station BULLETIN No. 272 RATE OF MILK SECRETION AS AFFECTED BY ADVANCE IN LACTATION AND GESTATION Correction of Yield, Within a Lactation Period, for Length of Record and for Pregnancy BY W. L. GAINES AND F. A. DAVIDSON URBANA, ILLINOIS, JANUARY, 1926 CONTENTS PAGE INTRODUCTION. 3 SOURCE OF DATA AND STATISTICAL TREATMENT 5 EQUATION OF THE LACTATION CURVE 7 RESULTS ... 10 Farrow Cows 10 Gestating Cows 13 Calf Carried Five Months or Less 13 Calf Carried More Than Five Months 14 Age and Productive Level 18 Fat Percentage 26 DISCUSSION 27 CORRECTION FACTORS FOR LENGTH OF RECORD 33 CORRECTION FACTORS FOR PREGNANCY 33 SUMMARY 34 LITERATURE CITED 36 Note. An extract from the material presented in this bulletin appeared in the January, 1926, number of the Journal of General Physiology: Gaines, W. L., and Davidson, F. A. The effect of advance in lactation and gestation on mammary activity. Jour. Gen. Physiol. 9, 325-332. 1926. RATE OF MILK SECRETION AS AFFECTED BY ADVANCE IN LACTATION AND GESTATION Correction of Yield, Within a Lactation Period, for Length of Record and for Pregnancy By W. L. GAINES, Chief in Milk Production, and F. A. DAVIDSON, First Assistant in Dairy Husbandry INTRODUCTION The practice of 'breeders of dairy cattle in officially testing their cows for production differs in several ways. Two such differences that appear in the published data of the Advanced Registry are the length of record and pregnancy. The length of record that is, the number of days covered by the record is of course a major factor in the amount of milk produced. Pregnancy, or the length of time a calf is carried by the cow during the production period, is also a factor affecting milk yield. In order properly to compare records in which these factors differ, it is necessary to know something of the quantitative relation between the length of the record and the amount of milk produced, and of the quantitative effect of pregnancy thruout its course upon milk yield. The total amount of milk yielded during any lactation period that is, from parturition to the end of lactation or until recurrence of parturition is greatly influenced by the length of time over which lac- tation extends. The length of lactation, in turn, is profoundly influenced by the service period (parturition to conception) . Ellinger 1 found in the Red Danish breed a high correlation between length of lactation and service period: r = .943 zh .005. This coefficient affords a numerical measure of the closeness of the relationship between gestation and the termination of lactation. A similar coefficient of correlation between total yield and length of lactation has not been derived, so far as we are aware, but Hammond and Sanders 2 found the correlation between service period and yield to be r = .33 .016. They derived the equa- tion y = 8500 4250e~' c ** as expressing the relation between ser- vice period and yield for the lactation (y = milk yield in pounds and x = service period in days). According to this equation the lactation yield would be 4,250 pounds if conception occurred at once, and would reach 8,500 pounds as a limit if breeding were indefinitely post- 4 BULLETIN No. 272 [January, poned. From this equation Hammond and Sanders have derived a set of correction factors using a 100-day service period as a base. It will be apparent that their correction is intended to measure the combined effect of length of record and pregnancy on the lactation yield. Their results are based on 1,410 records of Shorthorn cows, chiefly nonpedigree, obtained from a milk recording society in England. The great majority of the published records of cows in the United States are for a partial lactation period starting shortly after calving and continuing for tunes varying from 7 to 365 days. Effective breeding may occur a few days after parturition or may be so delayed that the calf is carried any time from 280 days (full term) to days during the record period. The length of the record and of pregnancy are two of the many variable factors which affect the magnitude of the published record of production. We expect a record for 200 days to be greater than that for 100 days, but not twice as great. In records of the same length we expect the production of the farrow cow to exceed that of the gestating cow to some extent. A more definite quantitative expression of these relations, if they are not too irregular, may permit a scheme of correction that will reduce the record to its equivalent under standard conditions. The relation between the 7-day and 365-day records of the Holstein breed has been studied by Yapp 3 and by Gowen and Gowen. 4 Yapp considered all ages together and found the correlation between the milk records for the two periods within the same lactation to be r = .702 .01. Gowen and Gowen find this coefficient reduced to about .6 when confined to narrow age limits. The latter authors have derived a complete series of prediction equations or correction formulae to express, for various ages, the quantitative relations between the two records. These formulae are based on a linear equation, y a + bx, where y is the yield to be expected from the record, x, which is known. We are concerned in this study more particularly with records of varying length in the same lactation, say from 200 to 365 days, during which the calf is carried varying lengths of tune. Of particular interest is the relation between the 305-day and 365-day records, since these periods constitute a common basis of distinction in the official testing of several of our dairy breeds. It is obvious that milk yield is the result of the rate of milk secre- tion. It is convenient to use the term lactation curve for either the curve representing the true theoretical rate of milk secretion or that rep- resenting the approximate rate of secretion shown by the monthly yields; the context and notation will indicate the usage. If an equation for the lactation curve can be determined, then the equation may be made the basis for estimating the yield for any portion of the period 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 5 covered. That the lactation curve is capable of simple mathematical expression adapted to the purpose at hand is indicated by various in- vestigations. Sturtevant, 5 in an early study of the decrease in milk yield with advance in lactation, reached the conclusion that ". . . . the natural falling off in milk for each month from calving, is about nine percent of the yield of the preceding month." His month is a thirty-day period and his conclusion is based on the records of 83 cows (45 Ayrshires, 3 Jerseys, 35 natives), for 210 lactations during the years 1866-1880, in a private herd. Later investigators have, in general, confirmed the applicability of this method of expressing the lactation curve. The above relation may raise the presumption that the rate of milk secretion is continually decreasing at a rate proportional to its value at the moment, since changes of such nature are of common occurrence in natural phenomena and lead to precisely such a relation as found by Sturtevant. Brody, Ragsdale, and Turner 6 have recently shown that the decline in milk yield which occurs with advance in lactation con- forms to the law of certain chemical reactions and may be expressed as "M t = M e~ ht where M t = milk production during any month, t; M is the theoretical value of the milk flow at the time of parturition; e and k have the usual meaning." Their equation is another way of expressing, in a more general and precise mathematical form, the con- clusion of Sturtevant. Regardless of the inferences drawn by these authors as to the chemical processes of milk secretion, it is evident from the data which they present that the equation gives a curve con- forming well with the observed monthly yields, at least within certain time^ limits.* SOURCE OF DATA AND STATISTICAL TREATMENT The data of the present paper are taken from the published records of the American Guernsey Cattle Club: namely, Vols. 33 and 34, and No. 1 of Vol. 35, of the Advanced Register. The published record in- cludes for each cow the date of birth, date of calving, date of effective service, and the milk and fat yields by calendar months. For the pur- pose of classification the printed pages of the record were removed and backed with sheets of gummed paper board. The sheets were then so cut as to give the record of each cow on a card of convenient size. The A significant deviation of the observed values from the theoretical during the first month or two of lactation is noted by the authors, and further treated on a chemi- cal basis in a later paper. 7 For simplicity and for the purpose of the present dis- cussion, this discrepancy may be ignored for the time being, to be mentioned again later (pages 31 and 32). 6 BULLETIN No. 272 [January, work of classification was thus facilitated and possible errors of tran- script were avoided. For the record of each cow additional data were computed and re- corded on the card, as follows: age of cow at calving (where necessary); time in days from calving to the beginning of the first full calendar- month record of milk and fat yield; and the time in days from calving to conception. All records in which the time from calving to the be- ginning of the first full calendar-month record exceeded 60 days, and all records in which essential data were lacking were discarded. A total of 4,522 records were used, including entries and re-entries. The records were then grouped with respect to the time of con- ception in days after calving, intervals of 30.5 days, 1-31 days, 32-61 days, 62-92 days, and so on, to a final group of farrow cows, that is, cows not bred at 366 days after calving, being used. Conception is re- garded as occuring at the middle of the intervals, that is, at .5 month after calving in the first group; at 1.5 months in the second; and so on. This classification was made in order to study the effect of pregnancy. In order to study the effect of advance in lactation, each of the above thirteen groups was further separated into two subgroups: the first subgroup (a) including those records in which 1 to 30 days elapsed between calving and the beginning of the first full calendar-month record; and the second subgroup (b) including those in which 31* to 60 days so elapsed. The average yield for subgroup (a) for the first full calendar-month is taken to represent the yield for a month, the mid- point of which is one month after calving. This gives the first observa- tion for the lactation curve. The second observation for the lactation curve (the yield for a month, the mid-point of which is two months after calving) is the average yield for the second calendar month of subgroup (a) combined with the first calendar month of subgroup (b) ; and so on to the eleventh observation for the lactation curve, which is the average of the eleventh calendar month of subgroup (a) combined with the tenth calendar month of subgroup (b). A possible twelfth observation represented by the eleventh calendar month of subgroup (b) was not used. The average yields have been computed by tabulating the data by months in the form of a correlation table, using class intervals of 5 pounds for fat and 100 pounds for milk. The data for milk and fat have been converted to a single expression representing the physiological equiva- lent of 4-percent milk on the basis of gross energy value and designat- ed fat-corrected milk, F.C.M., in accordance with the writers' ideas pre- viously presented. 8 The equation is F.C.M. = AM + 15F, where M is milk and F is fat, and all in the same unit of weight, that is, the pound in the present data. The fat-corrected milk is to be regarded as an esti- 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 7 mate of the energy yield in terms of natural 4-percent milk (one pound F.C.M. = one pound of 4-percent milk = 330.62 large calories). EQUATION OF THE LACTATION CURVE If the lactation curve can be expressed as a rate or velocity in the form of a differential equation, capable of integration, we have then the means of computing the area under the curve, or the yield between any time limits desired. As suggested in the introduction, it is purposed to use the equation: = ae~ kt (1) dt W in which y = yield in pounds; t = time in months (30.5 days) from dij calving; - represents the rate of yield in pounds per month; a is a con- CH stant representing the theoretical initial rate of yield in pounds per month; A; is a constant representing the rate of change per month (as dii proportional to -^) in the rate of yield per month, the minus sign indi- CLL eating that the change is toward smaller and smaller values; and e = 2.71828. a a The significance of the factor e in the present connection may be illustrated by the familiar case of money at simple and compound interest. Let a represent the original principal bearing interest at the rate of r percent per annum, and let k = r/100, that is, the rate of interest expressed as a decimal. At the end of t years the amount at simple interest is a X (1 + kt) that is, a+ akt; but if the interest is added to the principal every instant the amount at the end of I years is a times e to the power kt, that is, ae kt . In the time required for the principal to double at simple interest (kt = 1), it would increase 2.71828 fold at true compound interest. If the principal were decreasing in a similar manner, instead of becoming a times e kt , it would become a divided by e kt that is, ^, which may be written ae~ kt . Under the assumption that the rate of yield is continuously decreasing at a rate proportional to its value at the moment, equation (1) may be derived thus: Let y' = -~-;t\ien ^- = ky' where A; is the constant of proportionality. Multiplying by f- we have ^- = kdt, and integrating, log y' = kt + C. If a represents y y v the initial rate, that is, a = y' when t = 0, then C = log a and log - = kt. dij Passing from logarithms to exponentials, y' ,r ae~ kt . Oil The rate of milk secretion does not change from moment to moment between milkings at the same rate as required by equation (1) when applied to longer periods, but for the purpose in view, and the gross periods which we shall have to consider in the integrated expression, we may ignore the periodic fluctuations correlated with the occurrence of milking. 8 BULLETIN No. 272 [January, Integration of (1) gives: fdy = f ae~ kt dt and, yfS -.r*< + C (2) Since we are dealing only with yields following calving, y = when t = 0, and we may evaluate C by substituting y = and t = in (2), giving: Substituting this value of C in (2), y = | (1 - e-*<) (3) If we have the constants a and k of (1), then we can compute y (= yield) up to any time or between any time limits from (3). From the eleven observations for the lactation curve we have to determine values for a and k of (1). Clearly, the observations are to be taken as definite integrals of (1); the first observation corresponds to /l.S /2.5 the theoretical value I ae~ kt dt; the second, to I ae~ ht dt; and so on, Jo.S J 1.6 /11.5 ae~ kt dt. 0.5 The ratio between the theoretical yield for any month and that of the next preceding month which will satisfy equation (1) is a constant, e~ k . Let y m designate the yield for a month, and let time (i) be counted to the middle of the month concerned. 3 Consider any two consecutive months, y mi (where t = ri) and y ma (where t = n + 1) and n is any assigned value (from .5 up), then: y mi = | ["(?-*< - -5) _ g-Hn + .5) j ( 4 ) and, The ratio of the yield for any month to that of the next preceding month is then, from (5) and (4) : "In the equations of the lactation curves, which henceforth are expressed in terms of y m and t, it will be understood that t is reckoned to the middle of the month. 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 9 [e-Hn+.S) _ e -i(n+1.5) 1 _. _ U - J _ ^ (6)b "* ^[~ e -*(n- .5) _ g-(n+ .5) Equation (4) may be written : 4 . _ |/>.5Jt _ /> .5fc | /> in ym ' " k l e r and returning to the general form, y m = Ae~ kt (7) where, and transposing, = ^ e .6fe _ e -. 5 * (8) Equation (7) is readily applied to the observed values. Ae~ k corresponds to the first observation; Ae~ 2k , to the second; Ae~ 3k , to the third and so on. We may write in place of (7), logio y m = logio A kt logio e and in this form the equation is linear in logio y m and t; also in logio A and k. From a straight line fitted graphically or by the method of least squares to logio of the observed yields, values for A and k of (7) may be determined (see Running, 9 Formula V). The constant, k, of (1) is the same as that of (7), and a of (1) is derived from A of (7) thru equation (8). When k is small, as in the present data, A and a are practically equal; thus, when k = .05, a = .999898A. While equations (1) and (7) are by no means mathematically identical they may, for practical purposes, be used interchangeably to express the lactation curve (cf. page 4) outside the influence of pregnancy. A similar usage of equations (9) and (10) in describing the lactation curve where the influence of pregnancy must be recognized, is also roughly justified, as will appear below. In the case of gestating cows, equation (7) is not sufficient to describe the lactation curve after pregnancy becomes somewhat ad- b The constant percentage decrease from month to month, previously referred to in Sturtevant's work, corresponds roughly to the constant of proportionality, k, in ratio (6), and 100 k is approximately equal to the percentage decline. The approxi- mation holds only for small values of k. When k = .045, e~ k = .9560 and the per- centage decline is 4.40; whenfc = .090, e~ k = .9139 and the percentage decline is 8.61. 10 BULLETIN No. 272 [January, vanced. It is necessary to add a term to describe the decrease in yield associated with pregnancy. The equation used for cows in advanced pregnancy in its differential form is: ^ = ae~ kt - &e*-<> dt and as applied to the observed values, y m = Ae~ kt - Be *e-> (10) The first terms on the right in (9) and (10) are the same as used in (1) and (7) respectively. In the second terms, c is time in months from calving to conception and is determined, for the groups where used, as indicated under the description of the statistical classifications (page 6). Consequently, t c is time in months from conception. Equation (10) has been applied to the data of gestating cows in those groups where conception occured 5.5 months after calving or earlier. The constants A and k of the first term have been determined from the observations up to and including t c = 5.5 by the method above described. The constants B and K of the second term have been derived by a similar method from the deviations of the observed values from the calculated values of Ae~ kt , beyond t c = 5.5. It may be assumed that the deviations are due to pregnancy, and it will be apparent from the value given c that t c, as used in (10), measures time from conception to the middle of the month of pregnancy. Under these con- ditions, it may be shown that: (11) From the values found for the second term of (10), the second term of (9) may be derived by application of (11). It is evident we may substitute the time in months during which the calf is carried for t c of (9), and by integration of the second term alone compute the effect of pregnancy during any portion of the gestation period. RESULTS Farrow Cows.- In presenting the results for farrow cows, we may confine our attention to the observed and calculated data from (7), y m = Ae~ kt , bearing in mind that y m = yield for a month, and t = time in months from calving reckoned to the middle of the month under consideration. The numerical data for the farrow-cow group are given 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 11 in Table 1. Equation (7) has been fitted by the method of least squares to the three sets of observed data milk yield, fat yield, and fat-cor- rected milk yield. The purpose in presenting the three sets of data is to show the interrelation between them and the degree to which the cal- culated values conform in each case to the observed values. Since the curves have been fitted by the method of least squares, the root-mean- square errors give a comparative measure of the agreement between observed and calculated values for the three sets of data, if allowance is made for the relative magnitudes involved. Such an allowance can be fairly made by weighting the error for each curve by the reciprocal of its A. This has been done in the last line of Table 1. Considering the weighted error of the F.C.M. curve as 100, the error for the fat curve is accordingly 138 and that for milk, 147. The observed data and fitted curves of Table 1 are shown graph- ically in Fig. 1, the data for fat yield being multiplied by 25 to bring Fig 1- Date Of MM Sacretk otf cowj) Tim* In Months from Calring ft) them into approximation with the other data. A very good agreement is evident for all three, but in so far as equation (7) may represent the underlying law governing the change in rate of milk secretion with advance in lactation, it seems that the energy yield (F.C.M.) is more amenable than either milk yield or fat yield. It may be noted from Fig. 1 that the curve for milk yield declines most rapidly, that for fat yield least rapidly, while that for F.C.M. is intermediate. There is quite a marked progressive change in the composition of the milk of Guernsey cows with advance in lactation, as shown by the fat percentage data in Table 1. The F.C.M. values take account of the change in concentration of the fat and the accompanying solids-not-fat in the milk. From the equations in Table 1 it will be seen that the rapidity of decline of the curves in Fig. 1 varies with A;; the larger k, the more rapid the decline. 12 BULLETIN No. 272 [January, 02 & 2 K, _ H o q H! gw s & S- c O3 O ^ T3 f ^ -2 a "03 _ q'il H -rt T5 03 S 03 42 O bC a o o3 N m E 3 ^ 03 ^^51 ^ "c . 42 T3 03 --> 43 M - tf 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 13 The figures for milk yield in Table 1 confirm the results in the work of Brody et al 6 previously mentioned, as well as those presented in a later paper 10 in which the investigators show a value for A (as used here) of 1167.2, against 1167.6 as shown in Table 1; and of k, .0537 against .0523. Considering that the constants being compared are de- rived from different groups of Guernsey A. R. records, they are in sub- stantial agreement. The k's are directly comparable, but the A's per- haps are not, since Brody seems to have reckoned t at the end of the month, whereas it is reckoned above at the middle of the month. Gestating Cows. The milk yields, fat yields, and fat-corrected milk yields for the twelve groups of gestating cows are given in Table 2. Equations have been derived only for the F.C.M. values and the con- stants have been determined graphically by Running's 9 straight-line method, Formula V. The equations are given in Table 3. The calculated values from the equations and deviations of the observed values are given in Table 2. The average fat percentage is also given in Table 2. The fat percentage measures the relative rates of secretion of fat and of milk as a whole, and also affords an index of the relative rates of secretion of the several milk constituents. This significance of the fat percentage is mentioned to justify its inclusion with data on the rate of milk secre- tion. , The arrangement of Table 2 is such that comparisons may be made in two ways. To illustrate, the F.C.M. yields are given in line 5 of each group, and if the figures in line 5 for any group are followed across from left to right, they show the effect of advance in lactation, and after conception the combined effect of advance in lactation and gestation, for that particular group. On the other hand, if the figures of lines 5 are read from top to bottom of any column they show, after conception, the effect of the progress of pregnancy with the stage of lactation constant, but they involve different groups of cows. The heavy zigzag line in the table indicates where conception occurs, that is, the mid-point of the pregnancy group classification. The use of different groups of cows introduces an uncertainty on account of the variability in yield. If one reads, for example, the milk yields in the first month of lactation, Table 2, where gestation is not a factor, it is seen that the values are quite irregular from group to group. On this account, and in view of the regularity of the lactation curve that is shown by the farrow-cow group, it seems preferable to work sepa- rately with the data for each pregnancy group. Calf Carried Five Months or Less. We may deal with the first six pregnancy groups more or less collectively. They are those groups where 14 BULLETIN No. 272 [January, TABLE 2. AVERAGE RATE OF MILK SECRETION PER MONTH BY MONTHLY INTERVALS, WITH ADVANCE IN LACTATION AND GESTATION Advance in gestation Line 1 Advance in lactation month after calving (mid-point) 1 2 3 4 5 6 7 8 9 10 11 Cows conceiving 11. 5 months after calving I.. 2. . . 3... 4. .. 5... 6... 7... 162 1102 50 5 196 1046 48 7 196 972 46 7 196 918 44 8 196 860 43 7 196 824 42 4 196 790 40 8 196 752 39 4 194 727 38 1 193 692 37 2 193 662 36 1 4 58 4 66 4.80 4 88 5 08 5 15 5 16 5 24 5 24 5 38 5 45 1199 1183 16 1150 1137 13 1089 1093 -4 1040 1050 -10 1000 1010 -10 966 971 -5 927 933 -6 892 897 -5 863 862 1 834 829 5 804 796 8 Cows conceiving 10.5 months after calving 1.. 2... 3... 4... 5... 6... 7... 148 1057 48 7 172 1006 46 5 172 937 45 3 172 879 43 3 172 833 42 4 172 800 41 172 767 39 8 172 740 38 7 172 704 37 7 172 680 36 4 172 642 34 9 4 61 4 62 4 83 4 93 5 09 5 13 5 19 5.23 5.36 5 35 5 44 1153 1134 19 1100 1092 8 1054 1052 2 1001 1014 -13 969 977 -8 935 941 -6 904 907 -3 877 874 3 846 842 4 818 812 6 780 782 -2 Cows conceiving 9.5 months after calving 1.. 2... 3... 4. .. 5... 6... 7... 188 1135 51.3 235 1079 49.3 235 1009 47.9 235 939 45.9 235 882 44 5 234 839 43.0 234 797 41.3 234 754 39.5 233 725 38.4 233 691 37.1 233 660 35.1 4 52 4 57 4 75 4 89 5 05 5 13 5 18 5 24 5 30 5.37 5 32 1223 1218 5 1171 1166 5 1123 1117 6 1064 1070 -6 1021 1025 -4 980 982 -2 938 940 -2 895 901 -6 866 863 3 833 826 7 791 792 -1 Cows conceiving 8.5 months after calving 1.. 2... 3... 4... 5. .. 6... 7... 237 1060 50.4 4.75 1192 1191 1 266 1046 48.8 4.67 1150 1141 9 266 973 46.5 4.78 1087 1093 -6 266 905 44.8 4.95 1034 1048 -14 266 857 43.7 5.10 999 1004 -5 266 822 42.5 5.17 966 962 4 266 780 41.0 5.26 926 922 4 266 739 39.3 5.32 885 883 2 266 705 37.4 5.30 844 846 -2 265 672 35.9 5.34 808 811 -3 264 642 34.7 5.40 777 777 Cows conceiving 7.5 months after calving 1. . 2. .. 3. .. 4... 5. .. 6... 7... 312 1067 48.9 4.58 1160 1165 -5 372 1031 47.6 4.62 1127 1115 12 372 959 45.9 4.79 1072 1068 4 372 895 43.8 4.89 1015 1022 -7 372 845 42.5 5.03 976 979 -3 372 800 41.1 5.14 937 937 372 764 39.5 5.17 898 897 1 372 723 37.7 5.21 854 859 -5 372 693 36.3 5.24 821 822 -1 372 662 35.0 5.29 790 787 3 372 634 34.3 5.41 767 753 14 Cows conceiving 6.5 months after calving 1.. 2... 3... 4. .. 5... 6... 7. .. 487 1089 49.1 4.51 1173 1173 575 1042 48.2 4.63 1140 1125 15 575 966 46.1 4.77 1078 1078 575 904 44.4 4.91 1028 1033 -5 575 854 43.1 5.05 988 990 -2 575 814 41.7 5.12 950 949 1 575 768 39.8 5.18 905 910 -5 575 737 38.2 5.18 868 872 -4 574 703 37.1 5.28 837 836 1 574 667 35.6 5.34 802 802 571 628 34.4 5.48 768 768 1 Line 1, number of records averaged. Line 2, raw data of milk yield, in pounds. Line 3, raw data of fat yield, in pounds. Line 4, average fat percentage. Line 5, fat-corrected milk yield, in pounds, observed. Line 6, fat-corrected milk yield, in pounds calculated. Line 7, deviation of observed from calculated F.C.M. values, in pounds. effective breeding occurred 11.5, 10.5, 9.5, 8.5, 7.5, and 6.5 months after calving. The data are given in the first part of Table 2, and are shown graphically in Figs. 2 to 7. By study of the graphs, particularly, it is evident that equation (7) fits the observations very well, apparently just as well as in the case of the farrow cows. It appears, therefore, that for the first five months of the gestation period pregnancy does not appreciably affect the lactation curve. Calf Carried More Than Five Months. After the fetus has reached an age of five months, the lactation curve is modified more or less, and 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 15 TABLE 2. Concluded Advance in gestation Line Advance in lactation month after calving (mid-point) 1 2 3 4 5 6 7 8 9 10 11 Cows conceiving 5.5 months after calving 1.. 2... 3... 4. .. 5. .. 6... 7... 636 1081 49 2 754 1037 47.8 753 967 46.0 753 907 44.3 753 851 42.4 753 809 40.8 753 771 38.9 752 734 37.8 752 698 36.4 752 660 35.1 747 611 33.4 4.55 1170 1174 -4 4.61 1132 1123 9 4.76 1077 1074 3 4.88 1028 1027 1 7.42 977 982 -5 5.04 936 939 -3 5.05 893 898 -5 5.15 860 859 1 5.22 825 821 4 5.32 791 784 7 5.47 746 746 Cows conceiving 4.5 months after calving 1... 2... 3... 4... 5... 6... 7... 615 1064 48 2 746 1018 47.4 746 949 45.4 746 883 43.3 745 827 41.2 745 784 39.5 745 745 38.1 745 707 36.9 745 660 35.3 742 609 33.3 727 539 30.8 4.53 1148 1163 -15 4.66 1118 1108 10 4.78 1061 1055 6 4.90 1003 1005 -2 4.98 949 957 -8 5.04 906 910 -4 5.11 869 868 1 5.22 836 825 11 5.35 793 784 9 5.47 744 740 4 5.68 675 684 -9 Cows conceiving 3.5 months after calving I... 2... 3. .. 4. .. 5. .. 6... 7... 439 1075 49.3 4 59 555 1021 48.0 4 70 555 948 45.8 4.83 555 889 43.5 4 89 555 832 41.1 4.94 555 791 39.9 5 04 554 744 38.3 5 15 554 697 36.7 5 27 553 643 34.8 5 41 542 570 31.7 5 56 486 502 28.8 5 74 1169 1178 -9 1128 1119 9 1067 1063 4 1008 1010 -2 950 960 -10 914 912 2 872 865 7 829 820 9 779 774 5 704 719 -15 633 633 Cows conceiving 2.5 months after calving I.. 2... 3. .. 4. .. 5... 6... 7... 183 1110 51 235 1068 49 6 235 987 46.7 235 915 44 2 235 867 42 6 235 813 41 6 235 769 40 235 724 38 5 235 664 36 4 235 580 33 1 235 478 28 1 4.59 1209 1203 6 4.64 1171 1148 23 4.73 1095 1094 1 4.83 1029 1044 -15 4.91 986 995 -9 5.12 949 948 1 5.20 908 903 5 5.32 867 858 9 5.48 812 808 4 5.71 729 739 -10 5.88 613 612 1 Cows conceiving 1.5 months after calving 1.. 2... 3. .. 4... 5... 6... 7. .. 104 1120 50.5 4.51 1206 1192 14 132 1052 48.5 4.61 1148 1136 12 132 969 45.6 4.71 1072 1083 -11 132 908 43.7 4.81 1019 1032 -13 132 852 41.9 4.92 969 983 -14 132 812 41.0 5.05 940 936 4 132 768 39.8 5.18 904 889 15 132 702 37.4 5.33 842 837 5 132 614 33.9 5.52 754 766 -12 132 508 28.9 5.69 637 637 Cows conceiving .5 months after calving 1... 2... 3... 4. . . 5. .. 6... 7... 23 902 41.8 4.63 988 1061 -73 29 940 43.2 4.60 1024 1014 10 29 881 40.4 4.59 958 970 -12 29 812 39.6 4.88 919 926 -7 29 778 37.7 4.85 877 884 -7 29 750 37.3 4.97 860 842 18 29 664 34.6 5.21 785 795 -10 29 598 32.7 5.47 730 728 2 29 481 27.3 5.68 603 603 NOTE. The heavy zigzag line thru the table denotes approximately the time of conception. Data from left to right show the effect of advance in lactation, plus (to the right of the heavy line) the effect of advance in gestation for the same group of cows. Data from top to bottom show the effect of advance in gestation (below the heavy line) independent of stage of lactation, but involve different groups of cows. equation (10) is used in place of (7). The data are given in Tables 2 and 3, and shown graphically in Figs. 8 to 13. The groups represented in Figs. 11, 12, and 13 have carried the calf 8.5 months at the last observation and show a more extended effect of pregnancy on the lacta- tion curve than occurs in Figs. 8, 9, and 10. It will be noticed from Table 3 that the constant .K is the same in each of the six groups where equation (10) is used. The constant B is also the same in four of the six groups. This might be taken to indicate that the de- crease in yield due to pregnancy is a fixed quantity, measured by the term Be K(i ~ c) (B = .01206, K = 1.09861). Representing this decrease by i di (inhibition) and time in months of pregnancy by p, we have = be Kp , dp 16 BULLETIN No. 272 [January, Fig Z - Bat* Of MilK. Secret (ar/ts time of concrpikm) T/me In Months From Calving (t) 9 10 II c 1000 I - Sate Of Mi'lK Secretion + MarKi time of conception} Time In Months From i) 800 IZC 9- Kate Of MM Secret ion MarKl time of conception) 77m In Months From Calving (t) i& 10- Rat* Of Him Secreti ( Marf> iimt of conceftio Time. In Months From Catr'ng(tJ It seems desirable, however, to study somewhat further the con- stancy of the decrease in yield which is associated with pregnancy. It may be that the effect varies with the age or productive level of the cow. Age and Productive Level. The groups bred at 1.5 and 2.5 months after calving have each been divided into four age groups: (1) one and two years, (2) three and four years, (3) five and six years, and (4) seven years and over. The yields have been computed as before and equation 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 19 f- I Fig II -Gate Of ft>/K Jfcretion ( narki iima of conception) T;m In Month* fr CalYinj)(t) Time. In Month) Trom Celyin({) i /3-Pd/e Of Mi'lK Jecret, ( Marti Age 5jnd6 //-> ( M time of conception) Time In Month) from Cjlr,ng(t) f,$-n-K a te Of M,IK Secretior Co> A/je 7anJ XM/-J time of conception) Tine In Monthi fro 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 21 T 1 1 r f/^ IS- Sate Of MilK Jffcretioi Cowi Age I and 2 Yeirt (IMirKs time of concept: on) Time. In Month) Tr Fij 19- Sate Of MM Secret, 01 Cowi Age 3 and 4 X<-dr3 (* Marts time of conception) Tine. In Monthi Tr 10- Bite OfM'iK Secretion Cows AgeSand6yrar3 ftarKi time of conception) . In Monthi fro fig Zl- Rita Of MilK Secretio COM) t&t 7 end * Years time, of conception) Tim* In Month* from Ctlrin4(t) BULLETIN No. 272 [January, fe O } ^ o o SH t ^5 O fj O} 20 B O h o I - OS O O O 2 o CO 1C * i-H 1C 1C _J_ 1C CO C _1_ CO O O _1_ co cO i CO CD _|_ 5 OS oooseoTt* CO CO C ^H COCO | i-H(N . CO CO 1 * C I-H i-H IN i-H 0000 | 5 o E o 49 a 'o 00 ooc c o COOS O ^H T-l i-H O i-H ICOOSi-H os t^- _i_ * OOO00 | OS OS I OO i i-H i-H i-H i-H | 1-H i-H 1 ^ oo coco O I-H IN OS t^ ^iC(Nt- OS O tv CO II > 00 0000 | CO (M CO | 1-H OS i-H (N i-H 1 5*1 13 g"g 8S 00 C^ I s * ^C i-H O ^f CO Tt* 00 COIN OS C3 Q^ fH OS i-H OO CO os os _(_ Tf CO Tfl i-H i-H r 1 1 i-H i-H 1 00 CO Tt* IN (NCOCO I (N (N \ i-H i-H CO OO CO 1C co co _L 1-H 1-H I IM . o 7>fl 5n ^ yri >"> 1 c 1 "O l_ 1 "S ' tn ^* -H 1C COt- ^iccot- i-H 1C COt- i-H 1C CO l^. *H o 1 J2-2 . S-^ c > > a 14- E 2 03 2 o rt e < 1 < p 8 (N o co a 03 .11 ~o 13 73 03 "" O a c c Pi c3 ci 03 ^) 1-H CO 1C t^ .c 24 BULLETIN No. 272 [Jamiary, M T3 O CO CD O * 00 OO O iO COCO O I-H OOO 'S i-H i-H .010 ss "Sfe (N CS OS 1 1 O l-H COCO + 00 OO OS i-H CD CO 1 SSS2 C<) COCO- ^ 10 OS CO i-H o o -4-3 -< ^^ OS 00l> COi oooo 1 OOOO + c3 h-q o "o co" Q} g 1 idt-^eo ^ TH (M CO OS s 3 Q 00 l> OS O5_l_ COCO 1 0000 + IO OS 1> l-H - O O "* CD * 00 OS i-H oo co co i oooo 1 lococo _i_ OS OS i C^OO GO I OS OS ' M r* ^ -t^> aT--H . PS p-> g O CO I>CO 3 i-H * OOt^i-H OO i-H i-H 1 OSOST^ 1OOO OS 1 OSOS 1 i 1 i 1 IO CD o o~l~ al O * O .s t s S =i H ^ O i o^t-co ^ 2 W ^H _O 1 1 * 00 ^~4 OO OO CO i-H O CO-* i-H (M I> O CO "^ ^ H *-3 efl l>t>- 1 OS OS _j_ OO 1 1 1 1 I 1 i-H i-H 1 ^^ -u ^" O 2 ^ jg So 03 100 oooo *^i os os co lOrHIO-* ^H-* CO(N H "*^ M (S g 0) < t>. l-H l-H I 00 00 ' OO O i-H i-H O O I i-H i-H 1 5 00 O E *-< G si O C HO 1 1 CO s ii+ * CO T}< OO oo7 i-H i-H "H S S + 1^ s 02 * i-H i-H 1 l-H i-H i-H i-H 03 L Si 2J d sg OOO- OO CO CO _|_ i-H i-H ~ i-H i-H lOCDCOlM i i I-H ~r ^H COCO O CO CO _L I^J "o ft 8 d G O H CD O 1 i-H CO--H (M | . I t-l O tt> P G w i > CO '^H ^A 1 *H h^ \ H \ i-H IO CO t i-H IO CO t>- ^U3CO^ i-H IO COI> M O . 10 J? - i 1 : > CO 03 P ll > o - - S"3 2 C l3 E" 1 ( i M a s tl ^ <* 1 4 $ l a a; ofT to ^> ^> m G^3 1 j > . ao > i i CO r- _G 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 25 1 M 1-4 i 1 t I 1 CO CO ^ CO OOOO U5OO OOOO O I 1 ! 1 1 1 t^ co co co Sl-H O I-I OI-H O tN II U r^rt<^>>> >> I-H OO Ot 10 o CO to " 8.s o w H O OS ll H o o g 3 H ac^Hooic i li li-HOO OO5O5C35Ci COOOOtNOlOCO .. o^ '^ oo *^ os co C 1 ^ o^ *o o^ ^? r** >o co ^H os t^* COCOC^C^i-ti-Hi-HOOOOC5O5OiOiOOGO 26 BULLETIN No. 272 [January, The values of B, however, show considerable variation between the different age classes. The value of B seems to be more closely related to the productive level, as represented by A, than it is to the age of the cows composing the group. It may be noted also from Table 6 that the value of k tends to in- crease with age and yield; that is, the younger cows are more "per- sistent" than the older cows. Just how far this greater persistency is associated with younger age independent of yield, and how far with a lower absolute production, independent of age, the present treatment of the data does not distinguish. Fat Percentage. The change which takes place in fat percentage with advance in gestation independent of advance in lactation may be determined from Table 2 by reading vertically, that is, from one group to another. As between the several groups at the same stage of lactation and before breeding, there is very little difference in the average fat percentage. Direct reading from the table then offers a simple and satisfactory way of studying the changes in fat percentage associated with gestation. The data from columns 9, 10, and 11 of Table 2, repre- senting months, the mid-points of which are respectively 9, 10, and 11 months from calving, are represented in Fig. 22. There is an increasing Fig 22- Effect of Pregnancy on fat Percentage Stage of Lactation Constant S IS 2.5 3.S -4.5 5.5 *.S J.S 8-5 9.5 Time From Conception Month* (Mid-Vomti of Months) tendency for the fat percentage to rise with advance in gestation after the fetus reaches four months of age. There is no particular object in attempting to find an equation for this rise, but it appears to be an ex- ponential change. The fat percentage appears to be appreciably affected at a slightly earlier stage of gestation than is the case with yield. Here also, as seems to be quite generally the rule, when the rate of milk secretion as a whole is changing appreciably, the change in the rate of fat secretion tends to lag behind the change in the rate at which the other constituents, taken collectively, are secreted. 1926} MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 27 DISCUSSION From the numerical and graphic presentation preceding, it is evi- dent that the lactation curve, as represented by the average of Guernsey Advanced Registry records of farrow cows, and also of gestating cows up to the end of the fifth month of pregnancy, is very well represented dii by the differential equation (1), -- = ae~ kt , notation as given above at (page 7). In biological material as variable as milk yield, one would hardly expect to find better agreement than is shown between the observed monthly yields and the calculated values. The root-mean- square error of the observations for farrow cows (Table 1), is less than one-half percent of the average monthly yield. The maximum deviation at any observation after the first for gestating cows is 18 pounds from the calculated value of 842 pounds (Table 2). The larger deviations occur, as a rule, in the observations of the earlier part of the lactation. The two constants of the equation can be given definite significance in terms of commonly recognized characters. The constant a, represent- ing the initial rate of secretion per month, is descriptive of the pro- ductive ability of the cow at the first of lactation, that is, for a month period. On the other hand, the constant k, considered as a positive quantity, representing the rate of decrease per month in the rate of yield per month, is descriptive of that character commonly known as persistency. It is an inverse measure of persistency as the term is gen- erally known. The theoretical yield for the lactation period or any portion of it is a function of a and k For the entire lactation the theoretical yield of farrow cows is a/k. For a fixed value of k the yield for the lactation or any portion of it is proportional to a. For a fixed value of a the yield for the lactation is inversely proportional to k ; but for a partial lactation the relation is inverse but not strictly proportional. Brody, Ragsdale, and Turner have pointed out particularly that the form of the equation for the lactation curve is the form that also de- scribes the course of a monomolecular reaction; and they infer that the rate of milk secretion is controlled by some such chemical process. It seems clear from what we know of the nutrition of the mammary gland that, in the main, the elemental materials and energy required in the elaboration of milk in the gland are supplied more or less continuously from the food thru the blood. The assumption of the presence of some material in the nature of a catalyst which acts as a necessary inter- mediary in the chemical processes involved, is perhaps reasonable, as is also the assumption that the catalyst is very slowly and continuously destroyed in accordance with the equation. It would seem to be necessary 28 BULLETIN No. 272 [January, to assume that the hypothetical catalyst is confined to the milk-secreting cells, since, so far as the writers are aware, attempts to show the presence of an accelerating agent to milk secretion in the circulation of actively lactating animals have given negative results. Another possible explanation of the form of the lactation curve occurs in connection with the known calcium requirements of the lactating cow. The calcium outgo exceeds the calcium intake while lactation proceeds above a certain level. The depletion of the calcium reserves of the body might conceivably lead to a reduction in the rate of secretion in accordance with that found. At low yields calcium would not seem to be a possible limiting factor, but it is not certain that the rate of secretion continues to decline in conformity with the equation after a low value is reached. Equation (9), - = ae~ kt - be K(t ~ c} , (page 10), as applied thru at equation (10), serves to describe satisfactorily the lactation curve for cows in advanced gestation, as evidenced by the deviations given in Tables 2, 4, and 5 and by the graphic presentation in Figs. 8 to 21. The minus term of equation (9) represents the effect of pregnancy on yield. The values found for the constants (6 = .01147; K = 1.09861) give a result which is in accord with results of other investigators, namely, that pregnancy results in a constantly increasing reduction of yield, which, however, is scarcely appreciable during the first five months. The equation gives a total decrease for the first five months of 2.5 pounds, but the decrease for the entire period (9.2 months) is 256 pounds. The rate of decrease as represented by the equation is shown graphically in Fig. 23. Gowen, 11 by correlation methods applied to the records of Vol. 31 of the Guernsey Advanced Register, found a decrease in milk yield, due to carrying the calf 9 months, of 342 pounds in cows 3 to 3% years old; and of 628 pounds in cows 3^ to 4 years old. Assuming the milk to contain 5 percent fat, this would give F.C.M. values of 393 and 722 pounds respectively. Gowen's results are thus considerably higher than found above. His results are based on a linear relation between duration of pregnancy and reduction of milk yield. The present results indicate that the relation is distinctly non-linear. With reference to effect of pregnancy on composition of the milk, as indicated by the fat percentage, Gowen concludes that there is no influence. His conclusion is based on the correlation coefficient for fat percentage for the year and duration of pregnancy. The present results, however, show a quite pronounced increase in fat percentage accom- panying advance in gestation, independent of advance in lactation 1926] MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 29 (Fig. 22). This increase would, of course, be much less pronounced if merged in the average data for a year. The differences in the present results and those of Gowen must be due to the differences in method of treating the data, since the source of the data in both cases is essentially similar. A reason for the difference is not hard to find. As to the absolute amount of the decrease in yield associated with pregnancy, the partial correlation method used by Gowen takes account of the difference in level of production of the farrow cows and pregnant cows at the third month of lactation. It fails to take account of the difference between the several groups in rate of decline in yield as lactation advances. From Table 3 it may be observed that the pregnant cows tend to have a higher value of the k constant, that is, they decline more rapidly in yield than the farrow cows. There ap- pears to be no reason to attribute this difference to pregnancy. A further limitation of the correlation method is that its accuracy depends upon linearity of the regression, and this condition is not fully satisfied in the present case. As to the increase in fat percentage of the milk associated with advanced pregnancy shown in Fig. 22, a rough estimate indicates that the average fat percentage for the year would be increased by less than .1 in cows carrying the calf 9 months. For cows carrying the calf 5 months, the increase in the yearly fat percentage is negligible. Hence the coefficient of correlation between duration of pregnancy and average fat percentage for the year would be very low. Gowen found the coefficient to be .014 as an average of ten age classes. But it is not safe to conclude from this low coefficient that the two variables are not at all associated. The correlation coefficient as applied is simply inadequate to show the relation in sufficient detail in this particular case. In explanation of the decrease in yield with pregnancy two sug- gestions have been offered at various times one, that the decrease is due to the nutrients required by the fetus; the other that a hormone is produced and enters the circulation during pregnancy and acts as an inhibitor to milk secretion. Experimental evidence indicates that the net nutrients require- ment for gestation constitutes a very minor drain on the mother. Thus, Eckles, 12 as the result of careful experiments, found that pregnant cows on a maintenance ration for farrow cows finished gestation with fully nourished calves and with their own body weights unimpaired. If the decrease in milk yield during the course of pregnancy is due to the nutrients requirement of the fetus, then the curve of the rate of decrease might be expected to parallel the growth-weight curve of the fetus. Unfortunately accurate data on the growth of the bovine fetus 30 BULLETIN No. 272 [January, are not available. The equation of the first extra-uterine growth cycle of the Jersey female has been determined by Brody and Ragsdale. 13 There is no reason to suppose that the course of growth of the calf in the late stages of intra-uterine development should differ greatly from that immediately following birth. Robertson, 14 in fact, has presented indirect evidence from the data of Brody and Ragsdale that their equation represents also the late stages of intra-uterine growth. Robert- son has shown also that a similar relation exists in the growth of the infant. The data of Donaldson 21 and of Read 22 on the intra-uterine growth of the rat and the guinea pig, tend also to support the growth relations indicated above. For lack of better data we may use the ex- trapolated values of Brody and Ragsdale's equation in comparing the decrease in milk yield with the growth of the fetus. of Inhibition of retui X5 Time In Month) from Conception FIG. 23. SHOWING THE OBSERVED RATE OF INHIBITION AND THE PROBABLE WEIGHT OF THE FETUS WITH ADVANCE IN GESTATION The solid-line curve shows the rate of inhibition per month in the rate of yield per month with advance in gestation. The area under this curve represents the decrease in yield up to nine months due to pregnancy. The broken-line curve represents the probable weight of the fetus. The rate of inhibition does not parallel the rate of the probable draft of the fetus on the nutrients of the ma- ternal blood stream. It would seem that if the nutrients requirement of the fetus is the cause of the decrease in milk yield, the rate of decrease should be pro- portional to the weight of the fetus, on the assumption that the re- quirements of the fetus at any moment are proportional to its weight. This appears to be true in the case of the chick (cf. Needham, 23 Figs. 6 and 8). The two curves are given, therefore, in Fig. 23, the one to represent the weight of the fetus, the other to represent the rate of de- crease in yield due to pregnancy. The two curves are not parallel, and 1926] MILK SECRETION WITH ADVANCE IN LATCATION END GESTATION 31 it seems unlikely that the rate of milk secretion is affected at all pro- portionately to the nutrients required by the fetus. Lane-Claypon and Starling 15 found experimental evidence of a hormone of pregnancy which induces growth of the mammary gland and they suggested that the hormone acted also as an inhibitor to milk secretion. D'Errico 16 supports this inhibitor postulate on the basis of the results of blood transfusions from a pregnant to a lactat- ing bitch, showing a transitory decrease in the rate of milk secretion. Woodman and Hammond 17 report that the amount and character of the secretion of the mammary gland of heifers in first pregnancy undergoes a noticeable change in the fifth month. This is also the time at which the effect of pregnancy on milk secretion becomes apparent. A hormone of pregnancy inhibiting milk secretion and appearing about the fifth month seems to offer a plausible explanation of the decrease in rate of milk secretion in late gestation, and has been accepted by various investigators (cf . Hammond and Sanders, 2 Eckles 18 page 413, and Hooper 19 ). The rate at which such an inhibitor may be produced need not be proportional to the size of the fetus. Some rather meager data by Gaines 20 indicate that there is not only an inhibitor to milk secretion present in the blood of the pregnant goat, but that it persists for some time after parturition in the circulation of both mother and kid. This, together with results mentioned above, suggests the influence of pregnancy on the mammary gland presented diagrammatically in Fig. 24. The figure represents lactation as con- tinuous, that is, as without any dry period, and covers the last 2.5 months of gestation and the first 1.5 months of the lactation following. Teleologically one might say that gestation provides a mechanism which insures the development and preparation of the mammary gland for the secretion of milk for the post-natal nourishment of the young, and which inhibits the secretion, almost or entirely, preceding par- turition, and by the gradual removal of the inhibitor following partu- rition provides for some time an increasing milk supply to meet the increasing needs of the growing young. If it be assumed that the production of the inhibitor ceases at parturition and that the amount then present in the circulation is destroyed or eliminated at a rate proportional to its concentration at the moment, then the rate of milk secretion following parturition would be fltJ represented by the general equation = ae~ kt be~ klt , in which the last term represents the post-partum inhibition of pregnancy. Brody, Turner, and Ragsdale 7 have applied this equation, as representing the course of a consecutive chemical reaction, to observed milk yields 32 BOLLETIN No. 272 [January, immediately following parturition, and secured a very satisfactory fit. They have accordingly interpreted the rate of milk secretion for the first month or two of lactation as dependent on such a reaction or pro- cess. An alternative interpretation is suggested by the considerations presented in Fig. 24. 5 eoo Time In Months from FIG. 24. INFLUENCE OF PREGNANCY ON THE RATE OF MILK SECRETION Profound changes in the activity of the mammary gland occur during late pregnancy and following parturition. The lines F repre- sent projections of the lactation curve as found outside the influence of pregnancy, parturition recurring before lactation ceases. The hor- izontally shaded area represents the effect of pregnancy in promoting the growth or rejuvenescence of the gland with reference to its po- tential secreting capacity; this effect is pictured as abruptly termin- ated at parturition. The vertically shaded area represents the decrease in milk yield observed in gestating cows as compared with the ex- pected yield when unbred. The horizontally and vertically shaded areas together represent the inhibition of milk secretion preceding parturition. The persistence of this inhibition after parturition is rep- resented by the stipled area. The ordinates of the lower border of the shaded areas represent the rate of milk secretion. The ordinates of the upper border of the shaded areas represent the potential capacity of the gland as to rate of secretion. The shaded portions of the ordinates represent the rate of inhibition of the rate of milk secretion It is assumed in Fig. 24 that the hormone which inhibits milk secretion is separate from that which promotes growth and rejuven- escence of the gland. If the two are the same or if the latter also persists after parturition, it becomes necessary to modify somewhat the upper curve of Fig. 24 to take account of the influence after parturition. Under these conditions the sharp point of the curve would be rounded down. But since this modification is small in value as compared to the total 1926] MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 33 values where it applies, it would not materially affect the applicability of the above equation. CORRECTION FACTORS FOR LENGTH OF RECORD The lactation curve, that is the rate of milk secretion, may be dy expressed as = ae~ kt except as to certain complications associated at with pregnancy. Outside the influence of pregnancy, therefore, the equation of the lactation curve offers a basis for determining the quanti- tative relations between the yields for various periods of the lactation. If we represent the standard length of record by S and wish to secure the ratio of S to some other length of record, L, both records starting at the same time, n, after calving, the ratio is: 'n + S /" Jn ae~ kt dt , 1 e~ sk n+L ... ae~ kt dt From Table 3 the average value of A; is .04412. If time is reckoned in days instead of months, k becomes .04412/30.5 = .00144656. By using this value of k and 305 days as the standard length of record, the values shown in Table 7 are obtained from the above ratio. It should be borne in mind that the factors given in Table 7 are based on a constant value of k and their applicability is limited by this fact. According to the table, the 305-day yield is 86.97 percent of the 365-day yield; but if k becomes .1 (instead of .04412) the 305-day yield becomes 90.46 percent of the 365-day. Between individuals there is presumably some variability in the values of k. From Table 6 it seems that k varies also with age and productive level. The factors of Table 7 are offered, therefore, tenta- tively and as an approximation. CORRECTION FACTORS FOR PREGNANCY The correction factors for pregnancy are derived from the equations dii of the lactation curves for farrow and gestating cows: -r- = ae~ kl and at dy K _ . = ae~ kt be , respectively. Using the record of farrow cows for a period of ten months as the standard, the ratio of that record to the corresponding record of gestating cows is given by: M a-lOk\ J? f-l a-Wh\ ! U e ) J\. (L e ) _ 34 BULLETIN No. 272 [January, Since c = months from calving to conception, 10-c = months of preg- nancy. From values previously given, K = 1.09861 and k = .04412. From Table 6, b/a = .00001, approximately. By applying these values in the above ratio, the correction factors for pregnancy given in Table 8 have been derived. Since the correction for pregnancy is of small magnitude, it is per- missible first to correct the record for length of time, if different from 305 days, and then to apply the correction of Table 8 to take care of the time the calf is carried. TABLE 8. CORRECTION FACTORS FOR TIME CALF is CARRIED To Convert to 305-Day Farrow Basis P time in days that calf is carried, commencing at conception. P 1 2 3 4 5 6 7 8 9 20... 1.0015 1.0015 1.0016 1.0017 1.0017 1.0018 1.0019 1.0019 1.0020 1.0021 21 1.0022 1.0023 1.0024 1.0025 1.0025 1.0026 1.0027 1.0028 1.0029 1.0030 22 1.0031 1.0033 1.0034 1.0035 1.0036 1.0037 1 . 0039 1 . 0040 1.0042 1 . 0044 23 1.0045 1.0046 1.0048 1.0050 1.0052 1.0054 1.0056 1 . 0058 1.0060 1.0062 24 1.0065 1.0067 1.0069 1.0072 1.0075 1.0077 1.0080 1.0083 1.0086 1.0089 25 1.0093 1.0096 1.0100 1.0104 1.0107 1.0111 1.0115 1.0119 1.0124 1.0128 26 1.0133 1.0138 1.0144 1.0149 1.0155 1.0160 1.0166 1.0173 1.0179 1.0185 27 1.0192 1.0199 1 . 0207 1.0215 1 . 0223 1.0231 1 . 0240 1 . 0249 1.0258 1 . 0268 28 1 . 0278 SUMMARY The study is based on the calendar month records of 4,522 yearly records of the Guernsey Advanced Registry. The records have been handled statistically to secure the monthly yields for farrow cows and cows bred at monthly intervals following parturition. Yield has been measured in terms of 4-percent milk (F.C.M.) on a gross energy basis (one pound F.C.M. = 331 large calories), estimated as AM + 15F, (M = milk, F = fat, all in pounds). Equations have been derived expressing the rate of milk secretion dy of the general type -- = ae~ kt for farrow cows and cows carrying the at Sift , calf five months or less; and = ae~ kt be K(t ' for cows carry- ing the calf more than five months (y = yield, t = time in months from calving, and c = time in months from calving to conception) . In these equations a is representative of the initial level of production and k is representative of persistency of milk flow, being an inverse measure of this characteristic. The minus term of the second equation measures the effect of pregnancy on milk yield. With 305 days as the standard, the ratio of the 305-day record to that of records of other lengths was derived from the corresponding definite integrals of the first equation. On the basis of the average 1926\ MILK SECRETION WITH ADVANCE IN LACTATION AND GESTATION 35 value of k, .04412, and starting at the same time after calving, the ratios for records varying from 200 to 365 days in length have been computed and tabulated by intervals of one day. The 305-day record is thus equal to 86.97 percent of the 365-day record. The value of k is associated to some extent with age and productive level. Younger cows have a lower value of k than older cows, that is, the younger cows are more "persistent" milkers. The influence of pregnancy on yield is regarded as caused directly by a physiological inhibitor to milk secretion in the circulation, rather than as due indirectly to the draft of the growing fetus on the nutrients of the blood. The value found for K was very consistently 1.09861. The value of 6 was roughly proportional to a (b = .0000 la). On this basis the ratio of the 305-day record of farrow cows to the 305-day record of gestating cows has been computed and tabulated for lengths of pregnancy varying from 200 to 280 days, by intervals of one day. The average decrease in yield for the first 5 months of pregnancy, by the equation, is 2.5 pounds, and for the gestation period (9.2 months) 256 pounds F.C.M., or 85 therms. 36 BULLETIN No. 272 LITERATURE CITED 1. ELLINGER, T. U. The variation and inheritance of milk characters. Proc. Nat. Acad. Sci. 9, 4, 111-116. 1923. 2. HAMMOND, J., and SANDERS, H. G. Some factors affecting milk yield. Jour. Agr. Sci. 13, 74-119. 1923. 3. YAPP, W. W. A study of the relative reliability of official tests of dairy cows. 111. Agr. Exp. Sta. Bui. 215. 1919. 4. GOWEN, MARIE S., and GOWEN, JOHN W. Studies in milk secretion, XVII. Maine Agr. Exp. Sta. Bui. 306. 1922. 5. STURTEVANT, E. L. Influence of distance from calving on milk yield. Report N. Y. (Geneva) Agr. Exp. 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