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. [Illustration: 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) [Illustration: 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 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).