Under these conditions, there would be a lighter incidence of birds in the sample triangle than in the upright triangle beside it (Figure 11, Diagram III). Compensation can be made by deliberately scaling down the computed size of the sample area below its actual size. A procedure for doing this is explained in Figure 11. If it were applied to present alt.i.tudinal data, it would place the computational flight ceiling somewhere below 4000 feet. In arriving at the flight densities used in this paper, however, I have used an a.s.sumed ceiling of one mile. When the alt.i.tude factor is thus a.s.signed a value of 1, it disappears from the formula, simplifying computations. Until the true situation with respect to the vertical distribution of flight is better understood, it seems hardly worthwhile to sacrifice the convenience of this approximation to a rigorous interpretation of scanty data. This particular uncertainty, however, does not necessarily impair the a.n.a.lytical value of the computations. Provided that the vertical pattern of migration is more or less constant, flight densities still afford a sound basis for comparisons, wherever we a.s.sume the upper flight limits to be. Raising or lowering the flight ceiling merely increases or reduces all sample cones or triangles proportionately.

A more serious possibility is that the alt.i.tudinal pattern may vary according to time or place. This might upset comparisons. If the divergencies were severe enough and frequent enough, they could throw the study of flight densities into utter confusion.

This consideration of possible variation in the alt.i.tudinal pattern combines with accidents of sampling and the concessions to perfect accuracy, explained on pages 379-385, to give to small quant.i.ties of data an equivocal quality. As large-scale as the present survey is from one point of view, it is only a beginning. Years of intensive work and development leading to a vast acc.u.mulation of data must elapse before the preliminary indications yet discernible a.s.sume the status of proved principles. As a result, much of the discussion in 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 s.p.a.ce, so nearly independently of the subjective interpretations of the observer. The best a.s.surance of the essential soundness of the flight density computations lies in the coherent results and the orderly patterns that already emerge from the a.n.a.lyses 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 ill.u.s.trated 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 lat.i.tudes where the moon reaches a point high overhead.

[Ill.u.s.tration: FIG. 12. Facsimile of form used to record data in the field. One sheet of the actual observations obtained at Progreso, Yucatan, on April 24-25, 1948, is reproduced here. The remainder of this set of data, which is to be used throughout the demonstration of procedures, is shown in Table 1.]

[Transcription of Figure 12's Data]

ORIGINAL DATA SHEET

DATE 24-25 April 1948 LOCALITY Progreso, Yucatan

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 s.p.a.ce 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.

[Ill.u.s.tration: FIG. 13. The identification of coordinates.

These diagrams ill.u.s.trate how the moon may be envisioned as a clockface, constantly oriented with six o'clock nearest the horizon and completely independent of the rotation of the moon's topographic features.]

[Ill.u.s.tration: FIG. 14. The apparent pathways of the birds seen in one hour. The observations are those recorded in the 11:00-12:00 P. M. interval on April 24-25, 1948, at Progreso, Yucatan (see Table 1).]

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 s.p.a.ced points on the circ.u.mference 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 circ.u.mference 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 lat.i.tudes. 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 compa.s.s, 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 ident.i.ty. 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 coordinates 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 coordinate. 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, ill.u.s.trated 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 coordinates to the corresponding lines pa.s.sing through the center of the circle, as in Figure 15. To ill.u.s.trate, 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.

[Ill.u.s.tration: FIG. 15. Standard form for plotting the apparent paths of flight. On these diagrams the original coordinates, exemplified by Figure 14, have been moved to center. In practice the sector boundaries are drawn over the circles in red pencil, as shown by the white lines in Figure 19, making it possible to count the number of birds falling within each zone. These numbers are then tallied in the columns at the lower right of each hourly diagram.]

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

==============================+============================== 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 coordinates 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 compa.s.s. 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 quant.i.ties of data. A simpler procedure is to divide the compa.s.s 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 compa.s.s, 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.

[Ill.u.s.tration: FIG. 16. Standard sectors for designating flight trends. Each zone covers a span of 22-1/2. The N_{6} and N_{8}, the N_{5} and N_{7}, and their south complements, where usually few birds are represented, can be combined and identified as N_{6-8} and N_{5-7}, etc.]

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)

[Ill.u.s.tration: FIG. 17. The meaning of symbols used in the direction formula.]

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 compa.s.s 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).

[Ill.u.s.tration: FIG. 18. Form used in the computation of the zenith distance and azimuth of the moon.]

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"

ill.u.s.trates 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 Alt.i.tude 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 alt.i.tude 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 lat.i.tude 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 lat.i.tude of the observation station is in the northern hemisphere, the H.O. 214 tables ent.i.tled "Declinations Contrary Name to Lat.i.tude" are used with south declinations of the moon, and the tables "Declinations Same Name as Lat.i.tude," 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 lat.i.tude of Progreso, Yucatan 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 alt.i.tude, 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 alt.i.tude 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.