THE FLIGHT ACTIVITIES OF THE HONEYBEE 27 



ber. For this period of the day it may be assumed, 3 for the purpose 

 of estimating the duration of flight, that the two lines are parallel 

 and that the above formula is applicable. To reduce the experimen- 

 tal error arising from this assumption, a period of flight is selected 

 when the number of bees in the field at each reading in this period is 

 known to be very near a constant number. The average of the num- 

 ber of bees in the field at each of the readings in this period is taken 

 as the constant number (/) in the above formula. The number n 

 may be taken from either the outgoing or the incoming totals, thus 

 giving two answers for the duration of flight which may be used as 

 a check on each other. 



This method shows that on any day when the number of bees in 

 the field is large it naturally follows that the flight is long; or, con- 

 versely, whenever the duration of each trip is short the number of 

 bees in the field is relatively small. (Compare Tables 3 and 4.) 



Care must be exercised in the selection of the data to be used 

 in the determination of the average duration of each voyage, for 

 if there has been a large loss in the number of bees which went to 

 the field, the figures would at first glance suggest that the number 

 of bees in the field, instead of remaining constant, is continually 

 increasing, thus giving a larger figure for the average duration of 

 each trip. 



The data used for the determination of the average duration of 

 each trip have been arranged (Table 2) to show the figures used in 

 the calculations for each of the days given. 



The probable error of the average was determined according to 



Bessel's formula, 0.6745.- / — — — , and shows the precision of the 



\ n(n— 1) 



assumed constant number of bees in the field for each of the days 

 given. To facilitate comparisons the probable error of the average 

 has also been expressed as a percentage of the average. 



Since there is a direct ratio between the average duration of the 

 voyages and the average number of bees in the field during the period 

 under consideration, it naturally follows that the probable per- 

 centage error of the average duration of the voyages, given in the 

 last two columns of Table 2, is liable to the same probable percent- 

 age error as the average number of bees in the field. The remainder 

 of the table is self-explanatory. 



- The straight line method has been applied to the data for May 15, July 9, and 

 July 10, following the general formula — 



(I) 2n+2x(m)=7;y 

 (II) 2x(n)+ZxHm)=Zxy, 

 solving for m and n and substituting in y=n + mx, to determine the two points which 

 fix the straight line sought. 



For May 15, the straight line representing exits begins at number 4|,501 and ends 

 at 30,88:;, and gives 15.27 minutes as the average duration of the trips; the straight line 

 representing returns begins at 3,133 and ends at 35,111, and gives 15.40 minutes as the 

 average duration of the tripe. Pot July 9 the exit numbers are 0,480 and 17,908, and 

 the resulting average duration of trips 93.78 minutes ; the numbers for the returns 

 are 2,742 and 14,510, and the average duration of trips 91.20 minutes. For July 10 

 the exit numbers are 5,114 and 14,339, and the. average duration of trips 102.13 minutes; 

 the numbers for the returns 1,905 and 10,876, and the average duration, of trips 

 105.74 minutes. . , .. ., 



This method shows that the lines for the three days diverge somewhat as the time 

 progresses. The average duration of the trips calculated on these lines differs so 

 slightly from the figures obtained by the former method (see Table 2) that for the pur- 

 poses of this paper, where no dose comparisons in the average duration of the voyages 

 is attempted, the formula *-*-£ derived above has been applied directly to the original 

 data in Che compilation of Table 2. 



