Part II of this paper is speculative in intent, and most of the

conclusions suggested are of a provisional nature. Yet, compared with similar procedures in its field, flight density study is a highly objective method, and a relatively reliable one. In no other type of bird census has there ever been so near a certainty of recording _all_ of the individuals in a specified space, so nearly independently of the subjective interpretations of the observer. The best assurance of the essential soundness of the flight density computations lies in the coherent results and the orderly patterns that already emerge from the analyses presented in Part II.

B. OBSERVATIONAL PROCEDURE AND THE PROCESSING OF DATA

At least two people are required to operate an observation station--one to observe, the other to record the results. They should exchange duties every hour to avoid undue eye fatigue. Additional personnel are desirable so that the night can be divided into shifts.

Essential materials and equipment include: (1) a small telescope; (2) a tripod with pan-tilt or turret head and a mounting cradle; (3) data sheets similar to the one illustrated in Figure 12. Bausch and Lomb or Argus spotting scopes (19.5 ×) and astronomical telescopes up to 30- or 40-power are ideal. Instruments of higher magnification are subject to vibration, unless very firmly mounted, and lead to difficulties in following the progress of the moon, unless powered by clockwork. Cradles usually have to be devised. An adjustable lawn chair is an important factor in comfort in latitudes where the moon reaches a point high overhead.

[Transcription of Figure 12's Data]

ORIGINAL DATA SHEET

DATE 24-25 April 1948 LOCALITY Progreso, Yucatán

OBSERVERS Harold Harry; George H. Lowery

WEATHER Moderate to strong "trade" winds along coast, slightly N of E. Moon emerged above low cloud bank at 8:26.

INSTRUMENT B. & L. 19.5 Spotting Scope; image erect

REMARKS Observation station located 1 mile from land, over Gulf of Mexico, at end of new Progreso wharf

-----------+------+-------+---------------------------------------- TIME | IN | OUT | REMARKS -----------+------+-------+---------------------------------------- C.S.T | | | 8:26 | -- | -- | observations begin; H.H. observing 50 | 4:30 | 9 | slow; small 56 | 3 | 10 | medium size 9:00 | 2 | 10:30 | very small 11 | 5 | 9:30 | moderately fast 25 | 5 | 10 | very small; rather slow 26 | 3 | 11 | " " 36 | 5 | 10 | medium size 40 | 3 | 10 | " " 43 | 5:30 | 9 | " " 46 | 3:30 | 10 | small 56 | 4:30 | 10 | medium size 9:58-10:00 | -- | -- | time out to change observers; G.L. at 10:05 | 4:30 | 11:30 | scope small 06 | 3 | 11 | 12 | 5 | 8 | very small 25 | 5 | 12 | very fast; small 30 | 4 | 10 | small 32 | 4 | 11 | " 32 | 2 | 11 | " 33 | 5 | 11 | " 33 | 4 | 1 | " 33 | 5:30 | 11 | " 35 | 4:30 | 10 | swallow-like 36 | 5 | 1:30 |

As much detail as possible should be entered in the space provided at the top of the data sheet. Information on the weather should include temperature, description of cloud cover, if any, and the direction and apparent speed of surface winds. Care should be taken to specify whether the telescope used has an erect or inverted image. The entry under "Remarks" in the heading should describe the location of the observation station with respect to watercourses, habitations, and prominent terrain features.

The starting time is noted at the top of the "Time" column, and the observer begins the watch for birds. He must keep the disc of the moon under unrelenting scrutiny all the while he is at the telescope. When interruptions do occur as a result of changing positions with the recorder, re-adjustments of the telescope, or the disappearance of the moon behind clouds, the exact duration of the "time out" must be set down.

Whenever a bird is seen, the exact time must be noted, together with its apparent pathway on the moon. These apparent pathways can be designated in a simple manner. The observer envisions the disc of the moon as the face of a clock, with twelve equally spaced points on the circumference marking the hours (Figure 13). He calls the bottommost point 6 o'clock and the topmost, 12. The intervals in between are numbered accordingly. As this lunar clockface moves across the sky, it remains oriented in such a way that 6 o'clock continues to be the point nearest the horizon, unless the moon reaches a position directly overhead. Then, all points along the circumference are equidistant from the horizon, and the previous definition of clock values ceases to have meaning. This situation is rarely encountered in the northern hemisphere during the seasons of migration, except in extreme southern latitudes. It is one that has never actually been dealt with in the course of this study. But, should the problem arise, it would probably be feasible to orient the clock during this interval with respect to the points of the compass, calling the south point 6 o'clock.

When a bird appears in front of the moon, the observer identifies its entry and departure points along the rim of the moon with respect to the nearest half hour on the imaginary clock and informs the recorder. In the case of the bird shown in Figure 13, he would simply call out, "5 to 10:30." The recorder would enter "5" in the "In" column on the data sheet (see Figure 12) and 10:30 in the "Out" column. Other comment, offered by the observer and added in the remarks column, may concern the size of the image, its speed, distinctness, and possible identity. Any deviation of the pathway from a straight line should be described. This information has no bearing on subsequent mathematical procedure, except as it helps to eliminate objects other than birds from computation.

The first step in processing a set of data so obtained is to blue-pencil all entries that, judged by the accompanying remarks, relate to extraneous objects such as insects or bats. Next, horizontal lines are drawn across the data sheets marking the beginning and the end of each even hour of observation, as 8 P. M.-9 P. M., 9 P. M.-10 P. M., etc. The coördinates of the birds in each one-hour interval may now be plotted on separate diagrammatic clockfaces, just as they appeared on the moon. Tick marks are added to each line to indicate the number of birds occurring along the same coördinate. The slant of the tick marks distinguishes the points of departure from the points of entry. Figure 14 shows the plot for the 11 P. M.-12 P. M. observations reproduced in Table 1. The standard form, illustrated in Figure 15, includes four such diagrams.

Applying the self-evident principle that all pathways with the same slant represent the same direction, we may further consolidate the plots by shifting all coördinates to the corresponding lines passing through the center of the circle, as in Figure 15. To illustrate, the 6 to 8, 5 to 9, 3 to 11, and 2 to 12 pathways all combine on the 4 to 10 line. Experienced computers eliminate a step by directly plotting the pathways through center, using a transparent plastic straightedge ruled off in parallel lines.

TABLE 1.--Continuation of Data in Figure 12, Showing Time and Readings of Observations on 24-25 April 1948, Progreso, Yucatán

==============================+============================== Time In Out | Time In Out ------------------------------+------------------------------ 10:37-10:41 Time out | 11:15 8 9:30 10:45 5:30 10 | 11:16 4 11 6 9 | 5 9 5:30 10 | 11:17 5 11:30 10:46 6 8 | 11:18 5 12 3:30 11 | 6 11:30 5 12 | 11:19 5:30 11:30 10:47 3:15 1 | 11:20 6 10 6 8:30 | 3 12 5:45 11:45 | 5 12 5 10 | 11:21 5:45 11 10:48 6 9:45 | 5 11 10:50 5:30 11 | 11:23 5 12 10:51 4 11 | 11:25 5 10:30 10:52 4 2 | 6 11 5:30 11 | 6 12 10:53 5:30 11:30 | 11:27 6 10 5 11 | 11:28 6 11:30 10:55 5 12 | 5:30 12:30 5 11 | 11:29 6 11:30 10:56 6 10 | 4 12 10:58 4:30 11:30 | 6:30 10:30 5:45 11:45 | 6 11 10:59 6:30 10:30 | 11:30 3 10 11:00 3:30 12 | (2 birds at once) 6:30 11 | 11:31 5 10:30 (2 birds at once) | 5:30 10:30 11:03 6 11 | 11:32 6 11:30 11:04 3 12 | 11:33 7:30 9:30 5 12 | 4 10:30 11:05 6 10 | 6 11:30 5 11 | 8 9:30 11:06 6 10:30 | 11:35 7 10 11:07 3 10 | 4:30 1 11:08 6 11 | 11:38 6:30 11 11:10 7 9:30 | 11:40 5:30 12 11:11 5 9:15 | 11:42 4 2 11:13 5 12 | 5 12 11:14 6:30 10 | 6 10 5:30 1 | 4 2 4 12 | 5 12 ------------------------------+------------------------------

Table 1.--_Concluded_ ==============================+============================== Time In Out | Time In Out ------------------------------+------------------------------ 11:44 8 9:30 | 8 10:15 7 11 | 12:16 3:30 1:30 6 10 | 8 11 11:45 5 12 | 12:23 7 1:30 6 10:30 | 6 12:30 5:45 11 | 12:36 8 11 4 12 | 12:37 7:30 1 11:46 7 11 | 12:38 7 12:30 6 12 | 12:40 8 1 11:47 8 10 | 12:45 7:30 1 11:48 6 10 | 12:47 5:30 1 11:49 6:30 10:30 | 12:48 7 1 11:51 8 10 | 12:52 5:30 1:30 8 10 | 12:54-12:55 Time out 8 10 | 12:56 8 10:45 8 10 | 12:58 5:30 1:30 6 10 | 7 1:30 8 10 | 7 2 6 11 | 12:59 5 3 7 12 | 1:00-1:30 Time out 11:52 5 1 | 1:37 8 12 11:54 7 11 | 1:38 8 12 6 12:30 | 1:48 7 1 11:55 5 12 | 7 1 11:56 7 10 | 1:51 5:30 11 5 12 | 1:57 8 1 11:58 8 11 | 2:07 7 2 11:59 5:30 12 | 2:09 9 12 12:00-12:03 Time out | 2:10 8 1 12:03 5:30 11:30 | 2:17 9 12 12:04 8 11 | 2:21 6 2 12:07 6 12:30 | 2:30 5:30 3:15 7:30 1 | 2:32 8 2 12:08 5 10:30 | 2:46 7 1 12:09 5:30 1 | 3:36 9 2 7:30 2 | 3:39 8:30 2 12:10 6:30 12:45 | 3:45 6 4 12:13 8 11 | 3:55 9 2 12:14 7 1 | 4:00 8 3 12:15 7 12:30 | 4:03 9 2 7:15 1:30 | 4:30 Closed station ------------------------------+------------------------------

We now have a concise picture of the apparent pathways of all the birds recorded in each hour of observation. But the coördinates do not have the same meaning as readings of a horizontal clock on the earth's surface, placed in relation to the points of the compass. They are merely projections of the birds' courses. An equation is available for reversing the effect of projection and discovering the true directions of flight. This formula, requiring thirty-five separate computations for the pathways reproduced in Figure 12 alone, is far too-consuming for the handling of large quantities of data. A simpler procedure is to divide the compass into sectors and, with the aid of a reverse equation, to draw in the projected boundaries of these divisions on the circular diagrams of the moon. A standardized set of sectors, each 22-1/2° wide and bounded by points of the compass, has been evolved for this purpose. They are identified as shown in Figure 16. The zones north of the east-west line are known as the North, or N, Sectors, as N_{1}, N_{2}, N_{3}, etc. Each zone south of the east-west line bears the same number as the sector opposite, but is distinguished by the designation S.

Several methods may be used to find the projection of the sector boundaries on the plot diagrams of Figure 15. Time may be saved by reference to graphic tables, too lengthy for reproduction here, showing the projected reading in degrees for every boundary, at every position of the moon; and a mechanical device, designed by C. M. Arney, duplicating the conditions of the original projection, speeds up the work even further. Both methods are based on the principle of the following formula:

tan [theta] = tan ([eta] - [psi]) / cos Z_{0} (1)

The symbols have these meanings:

[theta] is the position angle of the sector boundary on the lunar clock, with positive values measured counterclockwise from 12 o'clock, negative angles clockwise (Figure 17A).

[eta] is the compass direction of the sector boundary expressed in degrees reckoned west from the south point (Figure 17B).

Z_{0} is the zenith distance of the moon's center midway through the hour of observation, that is, at the half hour. It represents the number of degrees of arc between the center of the moon and a point directly over the observer's head (Figure 17C).

[psi] is the azimuth of the moon midway through the hour of observation, measured from the south point, positive values to the west, negative values to the east (Figure 17D).

The angle [eta] for any sector boundary can be found immediately by measuring its position in the diagram (Figure 16). The form (Figure 18) for the "Computation of Zenith Distance and Azimuth of the Moon" illustrates the steps in calculating the values of Z_{0} and [psi]_{0}. From the American Air Almanac (Anonymous, 1945-1948), issued annually by the U. S. Naval Observatory in three volumes, each covering four months of the year, the Greenwich Hour Angle (GHA) and the declination of the moon may be obtained for any ten-minute interval of the date in question. The Local Hour Angle (LHA) of the observation station is determined by subtracting the longitude of the station from the GHA. Reference is then made to the "Tables of Computed Altitude and Azimuth," published by the U. S. Navy Department, Hydrographic Office (Anonymous, 1936-1941), and better known as the "H.O. 214," to locate the altitude and azimuth of the moon at the particular station for the middle of the hour during which the observations were made. The tables employ three variables--the latitude of the locality measured to the nearest degree, the LHA as determined above, and the declination of the moon measured to the nearest 30 minutes of arc. Interpolations can be made, but this exactness is not required. When the latitude of the observation station is in the northern hemisphere, the H.O. 214 tables entitled "Declinations Contrary Name to Latitude" are used with south declinations of the moon, and the tables "Declinations Same Name as Latitude," with north declinations. In the sample shown in Figure 15, the declination of the moon at 11:30 P. M., midway through the 11 to 12 o'clock interval, was S 20° 22´. Since the latitude of Progreso, Yucatán is N 21° 17´, the "Contrary Name" tables apply to this hour.

Because the H.O. 214 expresses the vertical position of the moon in terms of its altitude, instead of its zenith distance, a conversion is required. The former is the number of arc degrees from the horizon to the moon's center; therefore Z_{0} is readily obtained by subtracting the altitude from 90°. Moreover, the azimuth given in the H.O. 214 is measured on a 360° scale from the north point, whereas the azimuth used here ([psi]_{0}) is measured 180° in either direction from the south point, negative values to the east, positive values to the west. I have designated the azimuth of the tables as Az_{n} and obtained the desired azimuth ([psi]_{0}) by subtracting 180° from Az_{n}. The sign of [psi]_{0} may be either positive or negative, depending on whether or not the moon has reached its zenith and hence the meridian of the observer. When the GHA is greater than the local longitude (that is, the longitude of the observation station), the azimuth is positive. When the GHA is less than the local longitude, the azimuth is negative.

Locating the position of a particular sector boundary now becomes a mere matter of substituting the values in the equation (1) and reducing. The computation of the north point for 11 to 12 P. M. in the sample set of data will serve as an example. Since the north point reckoned west from the south point is 180°, its [eta] has a value of 180°.

tan [theta]_{Npt.} = tan (180° - [psi]_{0}) / cos Z_{0}

Substituting values of [psi]_{0} found on the form (Figure 18):

tan [theta]_{Npt.} = tan [180° - (-35°)] / cos 50° = tan 215° / cos 50° = .700 / .643 = 1.09

[theta]_{Npt.} = 47°28´

Four angles, one in each quadrant, have the same tangent value. Since, in processing spring data, we are dealing mainly with north sectors, it is convenient to choose the acute angle, in this instance 47° 28´. In doubtful cases, the value of the numerator of the equation (here 215°) applied as an angular measure from 6 o'clock will tell in which quadrant the projected boundary must fall. The fact that projection always draws the boundary closer to the 3-9 line serves as a further check on the computation.

In the same manner, the projected position angles of all the pertinent sector boundaries for a given hour may be calculated and plotted in red pencil with a protractor on the circular diagrams of Figure 15. To avoid confusion in lines, the zones are not portrayed in the black and white reproduction of the sample plot form. They are shown, however, in the shaded enlargement (Figure 19) of the 11 to 12 P. M. diagram. The number of birds recorded for each sector may be ascertained by counting the number of tally marks between each pair of boundary lines and the information may be entered in the columns provided in the plot form (Figure 15). We are now prepared to turn to the form for "Computations of Sector Densities" (Figure 20), which systematizes the solution of the following equation:

(220) 60/T (No. of Birds) (cos^2 Z_{0}) D = --------------------------------------- (2) (1 - sin^2 Z_{0} cos^2 [alpha])^0.5

Some of the symbols and factors, appearing here for the first time, require brief explanation. D stands for Sector Density. The constant, 220, is the reciprocal of the quotient of the angular diameter of the moon divided by 2. T is Time In, arrived at by subtracting the total number of minutes of time out, as noted for each hour on the original data sheets, from 60. "No. of Birds" is the number for the sector and hour in question as just determined on the plot form. The symbol [alpha] represents the angle between the mid-line of the sector and the azimuth line of the moon. The quantity is found by the equation:

[alpha] = 180° - [eta] + [psi]_{0} (3)

The symbol [eta] here represents the position of the mid-line of the sector expressed in terms of its 360° compass reading. This equation is illustrated in Figure 21. The values of [eta] for various zones are given in the upper right-hand corner of the form (Figure 20). The subsequent reductions of the equations, as they appear in the figure for four zones, are self-explanatory. The end result, representing the sector density, is entered in the rectangular box provided.

After all the sector densities have been computed, they are tabulated on a form for the "Summary of Sector Densities" (Figure 22). By totaling each vertical column, sums are obtained, expressing the Station Density or Station Magnitude for each hour.

An informative way of depicting the densities in each zone is to plot them as lines of thrust, as in Figure 23. Each sector is represented by the directional slant of its mid-line drawn to a length expressing the flight density per zone on some chosen scale, such as 100 birds per millimeter. Standard methods of vector analysis are then applied to find the vector resultant. This is done by considering the first two thrust lines as two sides of an imaginary parallelogram and using a drawing compass to draw intersecting arcs locating the position of the missing corner. In the same way, the third vector is combined with the invisible resultant whose distal end is represented by the intersection of the first two arcs. The process is repeated successively with each vector until all have been taken into consideration. The final intersection of arcs defines the length and slant of the Vector Resultant, whose magnitude expresses the Net Trend Density in terms of the original scale.

The final step in the processing of a set of observations is to plot on graph paper the nightly station density curve as illustrated by Figure 24.