THE FLIGHT ACTIVITIES OF THE HONEYBEE 27 
ber. For this jjeriod of the day it may be assumed,^ 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 figiires 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 clata 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. 
3 The straight line, method has been applied to the data for May 15, July 9, and 
July 10, following the general formula — 
(I) Xn+Xx(m)=Zy 
(II) 2i(n)+2x2(m)=Ziy, 
solving- for m- and n and substituting iu if=)i + mx, to determine the two points which 
fix the straight line sought. 
For May lo, the straight line representing exits begins at number 4^501 and ends 
at 36.883. and gives 15.27 minutes as the average duration of the trips ; the straight line 
represejiting returns begins at 3,133 and ends at 35,111. and gives 15.46 minutes as the 
average duration of the trips. For July 9 the exit numbers are 6,486 and 17,908, and 
the resulting avei-age 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 ai-e 5,114 and 14,339, and the avei-age duration of trips 102.13 minutes; 
the numbers for the returns 1,965 and 10,876, and tlie 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 metbod (see Table 2) that for the pur- 
poses of this paper, where no close comparisons in the average duration of the voyages 
is attempted, the formula 2: = ^ derived above has been applied directly to the original 
data in the compilation of Table 2. 
