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EXPERIMENTAL

A Biometrical Study of the Influence of Size of Brood Cell Upon the Size and Variability of the Honeybee (Apis mellifera L.) by Roy A. Grout, 1931

A. Purpose of Study

The purpose of this experiment is to scientifically study the increase in size and variability of the worker bee as influenced by its rearing in brood cells constructed by worker bees on artificial foundation having an enlarged cell base. Three different cell sizes were used in this experiment. The number of cells per square decimeter were, 857, 763 and 706. The foundation having 857 cells per square decimeter is the standard commercial size manufactured in the United States while the two latter sizes approximate that having 750 cells per square decimeter which has been manufactured since 1896 by Jos. Mees Sons of Herenthals, Belgium and that having 700 cells per square decimeter which the same firm has manufactured since 1927.

B. Methods and Materials.

The foundation used in the experiment was furnished by Dadant and Sons, of Hamilton, Illinois. The sizes furnished were 857 cells per square decimeter (standard size manufactured in the United States), 763 cells per square decimeter and 706 cells per square decimeter. The two latter sizes were selected by Mr. H. C. Dadant and are approximately the same sizes as those being placed on the market by foundation manufacturers in Belgium and parts of France. The cutting of special dies and the manufacturing of the foundation were personally superintended by Mr. Dadant in order that the resulting cell bases should be true hexagons. The foundation received with the first shipment contained 7 vertical wires embedded in each sheet of wax. In addition to the vertical wires 4 horizontal wires were placed in the frames and embedded in each sheet of wax by hand. Some trouble was experienced with this foundation due to its warping between the embedded wires in warm weather. In the second shipment the foundation contained 10 wires embedded in the vertical position, which, when placed in a frame wired with 4 horizontal wires, did not warp and resulted in perfect combs when drawn out by the bees.

Some combs were used which had been constructed from special foundation placed in certain colonies during the summer of 1929 by Dr. O. W. Park. To facilitate recognition and handling of the combs, the system used by Dr. Park in marking the frames was followed in this experiment. The frames containing the standard-size foundation, having 857 cells per square decimeter, were marked “A” and one notch was cut in the top-bar. Likewise, the frames containing the foundation having 763 cells per square decimeter were marked “B” and two notches were cut in the top-bar; while the frames containing foundation having 706 cells per square decimeter were marked “C” and three notches were cut in the top-bar.

Since it is a well established fact that under normal conditions bees will extend the side walls of the cell and construct a comb containing cells of the same diameter as the imprint of the cell base on the artificial foundation, no control of size of cell other than special foundation was exercised.

Frames containing all three sizes of foundation were placed in each of 23 colonies of the Iowa State College Apiary early in the summer of 1930. In general, two frames of each size were placed in each colony. Individual colony records were kept and the queens were marked by clipping the right wings of those reared in an even-numbered year and left wings of those reared in an odd-numbered year.

An effort was made to collect the bees upon emergence from all three sizes of cells in a single colony at approximately the same time and under the same conditions. For this purpose a chart was made whereby the daily emergence of the bees from each size of cell was kept for all of the 23 colonies. Each frame was caged in a Root Nucleus Introducing Cage a day or two before the time of emergence and a selected area of brood was covered with an additional small screen cage insuring that the emerging bees would have no access to any nectar or honey. During the honeyflow, it was often difficult to find an area of brood that did not contain some uncapped cells of nectar and honey, and bees were not collected from such combs.

Each sample collected from a brood comb contained at least 50 bees. During the summer of 1930, over 6000 bees were collected. During June of 1931, over 600 bees were collected. From these collections, approximately 3500 were selected as being most suitable for the experiment. The bees of this group were in sets of 150 bees consisting of three samples of 50 bees each taken from each of the three sizes of cells from the same colony, from the same mother and at approximately the same time.

After collecting each sample, the bees were slightly anesthetized, either with ether or calcium cyanide, and then killed by dropping into boiling water. This method of killing, as shown by Alpatov (3), caused the proboscis to be fully extended. The sample was then preserved in a 70% alcohol solution for further treatment.

The general plan of procedure for measuring the size of the individual bees of a sample consisted of the following treatment: (1) Determining the weight of the individual bee. (2) Dissecting the right fore wing, the third tergite, the fourth tergite and the proboscis of each individual bee. (3) Mounting these parts for measurement. (4) Measuring the parts.

Experiments showed that an individual bee taken from a 70% alcohol solution, dried for a few minutes on a filter paper and placed on a chemical balance lost weight faster than it could be accurately weighed. It was thought best, therefore, to take the individual dry weight of each bee. Further experiments were run which showed that, by removing the bees from the 70% alcohol solution, drying on filter paper for several minutes to remove excess preservative and placing the sample in a De Khotinsky Constant Temperature Oven Appliance at a constant temperature of 70 degrees centigrade for 48 hours, the individual bees of the sample no longer lost any appreciable weight. The sample was then placed in a desiccator containing concentrated sulfuric acid. Further experiments showed that after 72 hours the bees had become thoroughly dried and no appreciable loss of weight occurred.

The individual bees were then taken from the desiccator and weighed by means of an Eimer and Amend chemical balance accurate to 0.1 mg. A container of fresh calcium chloride was kept within the chemical balance at all times to dehydrate the contained atmosphere. It was also found that the repeated opening of the desiccator during a long series of weighings caused the individual bees to increase in weight. This necessitated the weighing of a test sample at intervals during an extended series of weighings to determine the average gain in weight of the individual bees. All weights given in this experiments have, therefore, been corrected for this factor.

After weighing, each bee was placed in a numbered vial containing tap water at room temperature and throughout the following treatment was recognized as a definite individual. After being in the water for 24 hours, the bees were soft enough for dissection. With the aid of a Spencer Binocular Microscope, containing a 3.5x ocular and a 55 mm. objective, and an ordinary dissecting set, the right fore wing, the third tergite, the fourth tergite and the proboscis of each bee were dissected. The dissected parts were then mounted directly upon numbered glass slides with Bueston’s medium* and cover glasses were applied.

*Bueston’s Medium for Mounting
Water …………………. 50 c.c.
Glycerine ……………… 20 c.c.
Gum Arabic …………… 40 gm.
Chloral Hydrate ……… 50 gm.
Dissolve Gum Arabic in water. When dissolved, add Chloral Hydrate.
When this is dissolved, add Glycerine. Filter.

All linear measurements were taken by a projection method. The numbered glass slide was placed in a Leitz Simple Micro-Projector in a vertical position and projected upon a movable screen attached to the opposite wall. Upon the face of the screen was a horizontal and vertical scale and the screen was so constructed that the entire face could be rotated around its center in a plane perpendicular to the line of projection. This feature greatly facilitated measuring the projected parts since the measuring scale could be turned to any desired angle at which the part to be measured might happen to lie. The projection measurement apparatus was arranged so that a glass Spencer stage micrometer, having a scale 2 mm. in length ruled to 0.01 mm., placed in the Micro-Projector gave a corresponding projection of 2 mm. magnified 127 times on the scale of the movable screen.

The apparatus was calibrated by this method before and at intervals during each long series of measurements. It was thus possible to read directly the exact measurement of the part in hundredths of a millimeter. However, for the sake of convenience and in order to eliminate any personal equation involved in the reading of actual measurements of the parts of the bee, a reading was taken at the beginning of the part and another at its end, the true measurement being the difference between the two readings. Plate 1 diagrammatically shows the measurements taken on the right fore wing, the third tergite and the fourth tergite. Following the system used by Michailov (43), the widths of the third and fourth tergites were combined and the summation of the two widths was used thruout the computation.

Plate 1. Diagram showing measurements of right fore wing and tergites 3 and 4.

Plate 2 diagrammatically shows the measurements of the proboscis. In this manner the length of the submentum, the length of the mentum and the length of the glossa were obtained, the summation of the three lengths being the length of the proboscis. In only one group of bees was the length of the second member of the labial palpi taken.

Plate 2. Diagram showing measurements taken of the proboscis. G = from tip of labellum to anterior part of mentum. M = length of mentum. S-M = length of submentum. L = length of 2nd member of labial palpi. Length of proboscis = G + M + S-M.

The computation of the statistics was accomplished by recording the values of the measurements of each individual bee on a Hollerith Electric Tabulating and Accounting Machine. From the summations obtained in this manner, the arithmetic means, standard deviations, correlation coefficients, regression equations and other statistical constants were computed with the aid of a Monroe Calculating Machine. All formulas and methods used in the above computations are given by Wallace and Snedecor (69) in their bulletin entitled “Correlation and Machine Calculation” as revised by Snedecor in 1931.

C. Presentation of Data.

1. The size of the worker bee as influenced by size of brood cell.

A study of the three sizes of cells used in this experiment and their relation to each other reveal the following data. To facilitate an understanding of this and following data the size of the cell contained in a comb having 857 cells per square decimeter will be designated as size of cell “A”; the size of cell contained in a comb having 763 cells per square decimeter will be designated as size of cell “B”; in a similar manner the size of cell contained in a comb having 706 cells per square decimeter will be designated as size of cell “C”. Between the sizes of cells “A” and “B” there is a reduction of 94 cells per square decimeter and between the “B” and “C” sizes there is a reduction of 57 cells per square decimeter, making a total reduction of 151 cells per square decimeter between the “A” and the “C” size. It was also thought advisable to investigate the increase of linear measurement of the cell and it was found that there was an increase of 5.98% in the diameter of the cells between the “A” and the “B” size, an increase of 3.96% between the “B” and the “C” sizes and an increase of 10.18% between the “A” and the “C” sizes of cells.

From the bees collected during the summer of 1930, data are presented in this thesis on the bees from three colonies. A sample containing at least fifty bees was collected from colony 25 from an “A” comb on August 21, 1930. One week later, on the 28th of August, two samples of bees were collected from a “B” and a “C” comb, respectively. The bees from colony 21 were collected within a period of two days, two samples being collected from an “A” and a “B” comb, respectively, on August 18, 1930, and a third sample from a “C” comb on August 20, 1930. The bees from colony 18 were collected over an extended period of time. One sample was taken from an “A” comb on August 30, 1930, another from a “C” comb on September 7, and the third from a “B” comb on September 23. The individual hive records of these three colonies show that the bees from each colony were the progeny of the same mothers.

From the samples of bees collected from colony 25, complete data were obtained on 44 bees of the sample from the “A” comb, 47 bees from the sample from the “B” comb and 45 bees from the sample from the “C” comb. Similarly, data are presented on 40 bees from the “A” comb from colony 21, 43 bees from the “B” comb and 45 bees from the “C” comb. In the case of colony 18, complete data were obtained on 41 bees from the “A” comb, 48 bees from the “B” comb and 50 bees from the “C” comb.

The influence of the increase in the size of the brood cells upon the size of various measurements taken on the parts of the individual worker bees of colony 25 is shown in Table 1. The measurements presented for comparison, given in column 1, are dry weight, length of right fore wing, width of right fore wing, sum of the widths of the third and the fourth tergites, length of proboscis, length of mentum, length of glossa and the sum of the lengths of the mentum and the glossa. In the second, fourth and sixth columns are given the arithmetic mean of each measurement and the standard deviation of the mean on the groups of bees from each of the three sizes of cells. In the third column are given the differences between the arithmetic means of the bees from size of cell “A” and the bees from size of cell “B” and the standard deviation of the mean difference. The differences between the means of the bees from size of cell “B” and the bees from size of cell “C” are presented with their standard deviations in column 5. Similarly, the differences between the means of bees from size of cell “A” and size of cell “C” are presented with their standard deviations in the seventh column.

The values of the mean differences that are statistically significant are starred. The test for significance was accomplished by dividing the mean difference by its standard deviation and comparing the resulting values with the corresponding “t” values given in Table 16 by Wallace and Snedecor (69).

All values of the mean differences between the 8 characters of bees from size of cell “A” and bees from size of cell “B” are statistically significant and are therefore greater than would be the case if the bees were selected at random from the same population. In a comparison of the values of the bees from size of cell “B” and size of cell “C” it is shown that, with the exception of length of mentum, all means of the bees from size of cell “A” and size of cell “C” differ significantly in all eight cases.

In Graph 2 is presented a frequency diagram of the character dry weight for each of the three samples of colony 25. An examination of the three curves, each representing the frequency distribution of the bees from one size of cell, shows that not only do the arithmetic means and peaks of the curves differ widely but also the distributions, since the curves scarcely overlap. There is also a difference in the types of the curves. While the curves representing the frequency distribution of the dry weight of bees from size of cell “A” and size of cell “B” are quite similar, the curve representing the dry weight of the bees from size of cell “C” shows a much more extensive distribution.

The frequency distribution of the character length of right fore wing is presented graphically in Graph 3. It is of interest to note that there is a trend in the peaks of the curves and the frequency distributions showing that not only do the means of the groups differ significantly but that the group as a whole tends to increase in size as the size of the cell, with which it is associated, increases.

The frequency distributions of the characters, width of right fore wing, sum of the widths of the third and the fourth tergites and the length of proboscis are presented graphically in Graphs 4, 5 and 6, respectively. The curves of Graphs 4 and 5 show that there is a trend both in the peaks of the curves and in the curves themselves toward a larger value of the measurement of the character with an increase in the size of the cell from which the sample is taken. The curves represented in Graph 6 do not show the familiar trend between the peaks of the two curves representing the length of proboscis of bees from size of cell “A” and size of cell “B”. There is, however, a distinct difference between the peaks of the curves representing this character for bees from size of cell “B” and size of cell “C”. The trend of the distribution curves in all three cases is from a smaller to a larger length of proboscis as the size of cell is increased.

It is also of interest to investigate the percent increase of the arithmetic means of the various measurements of the bees as the size of the cell is increased. Dry weight, which is a measurement of mass and consequently volume, increases markedly from 15.50% to 51.28%, as the size of cell increases. The linear measurements show an average increase of 1.37% as the area of the cell base is increased 12.32%, an average increase of 1.00% as the area of the cell base is increased 8.07% and an average increase of 2.38% as the area of the cell is increased 21.39%. By comparing the average percent increase of the linear measurements to the increase of the diameter of the cell it is discovered that as the diameter of the cell is increased 5.98% there is an increase of 1.37% in the average linear measurement, that, with an increase of 3.96% in the diameter of the cell there is an average increase of 1.00% in the linear measurements and that an increase of 10.18% in the diameter of the cell is accompanied by a corresponding increase of 2.38% in the average measurements of the worker bees. It is of interest to note here that the ratios of the percent increases of the diameters of the cells are approximately the same as the ratios of the percent increases of the average dimensions of the worker bees.

The percent increase of the linear measurements on the various parts of the bees are shown diagrammatically in Graph 1 and the percent increase of all measurements is given below in tabular form.

Measurement Taken Percent Increase from “A” to “B” Percent Increase from “B” to “C” Percent Increase from “A” to “C”
Dry weight 15.50% 30.98% 51.27%
Length of right fore wing 0.60 0.89 1.49
Width of right fore wing 1.05 1.15 2.21
Sum of widths of third and fourth tergites 2.24 1.45 3.72
Length of proboscis 0.93 1.13 2.07
Length of mentum 1.49 0.11 1.61
Length of glossa 1.21 1.22 2.45
Sum of lengths of mentum and glossa 1.28 0.90 2.19
Average of seven linear measurements 1.37 1.00 2.38

The influence of the increase in the size of the brood cells upon the size of the worker bees from colony 18 is shown in Table 2. The data presented in this table consist of the arithmetic mean and the standard deviation of the arithmetic mean of the bees from the three sizes of cells and the mean difference and the standard deviation of the mean difference between each of the three groups. The measurements presented in the table are dry weight, length of right fore wing, width of right fore wing, sum of the width of the third and fourth tergites and length of proboscis. The mean differences which are statistically significant are starred.

An examination of Table 2 shows that between the means of the measurements taken on the bees from size of cell “A” and size of cell “B” there is only one case where the difference is significant. This is in the case of dry weight and in a negative direction. Table 2 further shows that there is a decrease in the length of the right fore wing between size of cell “A” and size of cell “B”, but the mean difference in this case is not significant. The other three measurements, width of right fore wing, sum of the widths of the third and the fourth tergites and length of proboscis show an increase in their respective means but the mean difference is not significant.

An examination of the arithmetic means of bees from size of cell “B” and size of cell “C” shows that the mean differences between the three groups are significant with the exception of the width of the right fore wing. In the case of the right fore wing there is an increase between the “B” and “C” groups but the increase is not statistically significant. An examination of the arithmetic means of the bees from size of cell “A” and size of cell “C” tells a similar story. The mean differences of all measurements are significant except for the measurement of the width of the right fore wing, whose means, while showing an increase from size of cell “A” to size of cell “C”, do not show a significant mean difference.

The influence of the size of the brood cell upon the size of various measurements taken on the bees from colony 21 is shown by a comparison of the arithmetic means of the measurements of the bees from size of cell “A”, size of cell “B” and size of cell “C”, respectively, in Table 3. Data presented in this table includes the erithmetic mean and it’s standard deviation for each measurement of the bees from each size of cell and the mean difference and its standard deviation for each measurement between the means of the bees of each group. The measurements taken on the parts of the bees of colony 21 and presented in Table 3 are dry weight, length of the right fore wing, width of the right fore wing, sum of the widths of the third and fourth targites and length of proboscis. The mean differences which are statistically significent are starred.

An examination of Table 3 shows that the mean differences for the measurements between the bees from size of cell “A” and size of cell “B” are all significant except for dry weight. There is a decrease in the dry weight of the bees of these samples as the size of the cell is increased, but the decrease is not large enough to be significant. There are no significant mean differences between the bees from size of cell “B” and size of cell “C” although all measurements, except the width of the right fore wing, show an increase as the size of cell is increased. An examination of the mean differences of the measurements on bees from size of cell “A” and size of cell “C” show that in all measurements, except dry weight, the mean differences are significant and that in all measurements there is an increase accompanying the increase in the size of brood cell.

2. The variability of the worker bee as influenced by size of brood cell.

In Table 4 are presented the correlation coefficients of measurements taken on the parts of the worker bees from the three sizes of cells from colony 25. The measurements taken upon the individual bees are given in column 1. Sizes of cell is designated in the second column. In the third column are presented the correlation coefficients of length of right fore wing with dry weight for each size of cell. The correlation coefficients of width of right fore wing with dry weight and width of right fore wing with length of right fore wing, for each of the three sizes of cells, are given in column 4. The correlation coefficients of the sum of the widths of the third and the fourth tergites with dry weight, with length of right fore wing and with width of right fore wing for each size of cell are given in column 5. Similarly, the correlation coefficients of length of proboscis, length of glossa, length of mentum, and the sum of the lengths of the mentum and the glossa with corresponding measurements in column 1 are given in columns 6, 7, 8 and 9, respectively, for the bees from each of the three sizes of cells. Those values which are starred with one star are highly significant correlations, while those values which are starred with two stars are significant correlations but not highly so. The values which are not starred have failed to meet the requirements of significance. Significance of the correlation coefficients was determined by comparing the values obtained with significant values of “r” given in Table 16 by Wallace and Snedecor (69).

Concerning the data presented in Table 4, the following assertions can be made: (1) The length of the right fore wing is significantly correlated with dry weight for the bees from all three sizes of cells, but only in the case of bees from size of cell “A” is the correlation highly significant. (2) Dry weight is highly significantly correlated with width of right fore wing in the case of bees from size of cell “A” and size of cell “B”, while in the case of bees from size of cell “C” the correlation coefficient approaches significance. (3) The correlation coefficient of the sum of the widths of the third and the fourth tergites with dry weight is highly significant in the case of the bees from size of cell “A”, is significant but not highly so in the case of the bees from size “B”. (4) The correlation of dry weight with length of proboscis, length of glossa, length of mentum, and the sum of the lengths of the mentum and the glossa is significant only in the case of bees from size of cell “A”. (5) The correlation of length of right fore wing with width of right fore wing is highly significant. (6) The correlation of length of right fore wing with the sum of the widths of the third and the fourth tergites is significant, but not highly so in the case of the bees from the size of cell “A” and size of cell “C”. The correlation coefficient in the case of bees from size of cell “B” is not significant. (7) A study of the correlation coefficients of length of right fore wing with length of proboscis and its integral parts shows a tendency for the correlation coefficients to be highly significant in the case of bees from all three sizes of cells. (8) There is no significant correlation between width of right fore wing and the sum of the widths of the third and the fourth tergites. (9) Concerning the correlation of the width of right fore wing with length of proboscis and its integral parts, there is a tendency for the correlation coefficient to be highly significant in the case of bees from size of cell “A”. In the case of the bees from size of cell “B” the tendency is for the correlation coefficient not to be significant and in the case of the bees from size of cell “C” the tendency is for the correlation coefficient to be significant but not highly so. (10) The correlation of the sum of the widths of the third and the fourth tergites with length of proboscis and its integral parts is not significant. (11) As would be expected, the correlation of the length of proboscis with length of glossa, length of mentum and the sum of the lengths of the mentum and the glossa is highly significant. (12) Concerning the correlation between length of glossa and length is not significant. (13) As would be expected, the correlation of the sum of the lengths of the mentum and the glossa with its integral parts, namely, length of mentum and length of glossa, is highly significant.

Data are presented in Table 5 concerning the correlation coefficients of measurements on bees from colony 21. The arrangement of this table is the same as that of Table 4. The measurements which have been correlated are dry weight, length of right fore wing, width of right fore wing, sum of the widths of the third and the fourth tergites, and length of proboscis. Similar to Table 4, those values which are highly significant correlations are starred with one star, those values which are significant but not highly so are starred with two stars, while those values which are unstarred are not significant.

Concerning the data presented in Table 5, the following general assertions can be made: (1) In all three cases the correlations of dry weight with length of right fore wing, dry weight with the sum of the widths of the third and the fourth tergites and dry weight with length of proboscis are significant. (2) The correlations of length of right fore wing with width of right fore wing and length of right fore wing with the sum of the widths of the third and the fourth tergites are significant. (3) The correlation of width of right fore wing with length of proboscis is significant. (4) The following correlations are not significant: dry weight with width of right fore wing, length of right fore wing with length of proboscis, width for right fore wing with the sum of the widths of the third and fourth tergites and the sum of the width of the third and the fourth tergites with length of proboscis.

In Table 6 are presented the correlation coefficients of measurements on bees from colony 18. The arrangement of this table is similar to table 4. The measurements upon which correlations have been calculated are dry weight, length of right fore wing, width of right fore wing, sum of the widths of the third and the fourth tergites and length of proboscis. The system for indicating the significance of correlation coefficients is the same as that used in Tables 4 and 5.

Concerning the data presented in Table 6, the following general assertions can be made: (1) The correlation coefficients of dry weight with length of right fore wing, dry weight with the sum of the widths of the third and the fourth tergites, length of right fore wing with width of right fore wing and width of right fore wing with length of proboscis are significant. (2) The correlation coefficients of dry weight with width of right fore wing, dry weight with length of proboscis, length of right fore wing with the sum of the widths of the third and the fourth tergites, length of right fore wing with length of proboscis, width of right fore wing with the sum of the widths of the third and the fourth tergites and the sum of the widths of the third and the fourth tergites with length of proboscis are not significant.

From the data presented in Tables 1, 2, 3, 4, 5 and 6, it is evident that the samples of bees from colony 25 are more homogeneous than the samples of bees from colony 21 and colony 18. Consequently, a further study of the variability of the worker bee as influenced by size of brood cell will be concerned with the bees from colony 25.

Throughout the following presentation of data, the measurements on the parts of the bee will be designated as follows: (A) dry weight, (B) length of right foe wing, (C) width of right fore wing, (D) sum of the widths of the third and the fourth tergites and (X) the measurement upon which the regression is made.

Throughout the following presentation of data, the measurements on the parts of the bee will be designated as follows: (A) dry weight, (B) length of right foe wing, (C) width of right fore wing, (D) sum of the widths of the third and the fourth tergites and (X) the measurement upon which the regression is made.

In Table 7 are presented further data concerning the measurements of bees from colony 25 in which there is a regression of (A) dry weight, (B) length of right fore wing, (C) width of right fore wing and (D) sum of the widths of the third and the fourth tergites on (X) length of proboscis. The data consist of the standard deviations of the above-mentioned measurements, the standard regression coefficients, the multiple correlation coefficient, the standard error of estimate, the significance of regression and the regression equations for the bees from each size of cell. An analysis of variance of the length of proboscis between and within all three groups, namely, the groups of bees from size of cell “A”, the group of bees from size of cell “B” and the group of bees from size of cell “C”, is also presented.

An examination of the standard deviations of the measurements of the bees from the three sizes of cells shows that the variation is greatest in the case of length of right fore wing and length of proboscis with the bees from size of cell “A” and least in the case of the bees from size of cell “B”. The standard deviation of the width of the right fore wing is greatest in the case of the bees from size of cell “A” and the least in the case of bees from size of cell “C”. The standard deviation of dry weight increases as the size of the cell is increased. The standard deviation of the sum of the widths of the third and the fourth tergites is greatest in the case of the bees from the size of cell “B” and least in the case of bees from the size of cell “A”.

The significance of the standard regression coefficients was tested by dividing the standard regression coefficient by its standard deviation and comparing the resulting value with the significant values for “t” as given in Table 16 by Wallace and Snedecor (69). The significance of the multiple correlation coefficients was determined by comparing the values obtained with significant values of “R” given by Wallace and Snedecor (69) in Table 16. The significance of the regression for the bees from each size of cell and the analysis of variance of length of proboscis of all three groups was tested by calculating one-half the difference of the natural logarithms of the mean squares and comparing the values obtained with the significant values of “Z” as given in Table 6 by Fisher*.

An examination of the standard regression coefficients shows that only the standard regression coefficint of length of proboscis and length of right fore wing is significant for the bees from each of the three sizes of cells. The multiple correlation coefficient for the bees from size of cell “A” and size of cell “C” are highly significant, while the corresponding value for the bees from size of cell “B” is not significant. The “Z” test of the significance of the regressions further substantiates the significance of the multiple correlation coefficients by showing that the regressions of the measurements on the bees from size of cell “A” and size of cell “C” has been significantly accounted for.

A study of the standard errors of estimate shows that the standard deviation of the length of proboscis of bees from size of cell “A” has been reduced 29.61% due to the extension of statistical control over factors relating to length of proboscis. In the case of the bees from size of cell “B” the standard deviation of length of proboscis has only been reduced 2.11%, while in the case of bees from size of cell “C” the reduction is 19.86%. In the latter case, the standard deviation of length of proboscis has been notably reduced by the inclusion of dry weight, length of right fore wing, width of right fore wing and the sum of the widths of the third and the fourth tergites in the regression. A study of the regression equations for the bees from the three sizes of cells shows, in general, that length of the right fore wing is the dominating factor in these estimation equations of length of proboscis.

An analysis of the variance between and within the groups of bees from all three sizes of cells shows that the variation between the groups is significantly greater that that within groups. This further substantiates the proof presented under section 1 of the presentation of data of a significant difference between the means of the length of proboscis of the three groups.

In Table 8 are presented statistical constants of measurements on bees from colony 25 concerning a regression of (A) dry weight, (B) length of right fore wing, (C) width of right fore wing and (D) sum of the widths of the third and the fourth tergites on (X) length of mentum.

In Table 9 are presented data concerning the statistical constants of measurements on bees from colony 25. In this table the regression is dry weight, length of right fore wing, width of right fore wing and sum of the widths of the third and the fourth tergites on length of glossa.

Statistical constants of measurements on bees from colony 25 concerning a regression of dry weight, length of right fore wing, width of right fore wing and sum of the widths of the third and the fourth tergites on the sum of the lengths of the mentum and the glossa are presented in Table 10.

A comparison of Tables 7, 8, 9 and 10 shows that the data presented in Tables 9 and 10 proffer the same conclusions as were drawn from the data of Table 7. From the data presented in Table 8, there is an agreement with the data of the other three tables in the variation of the length of mentum as indicated by the standard deviation of length of mentum of the bees of all three sizes of cells, the multiple correlation coefficient and the significance of the regression of the bees from size of cell “A”, and the analysis of variance of length of mentum between and within the groups from all three sizes of cells. In no cases are the standard regression coefficients significant. In contrast to the data presented in the other three tables, the multiple correlation coefficients and significance of regression for the bees from the size of cell “C” are not significant. An examination of the regression equations shows that length of right fore wing has ceased to be a dominating factor in the estimation of length of proboscis.

*Fisher, R. A., “Statistical Methods for Reasearch Workers”, second edition revised and enlarged. Oliver and Boyd, Edinburgh. 1928.