ADB  , the

plane of the horizon; EDF, the Z path of the sun . Let the circles ADB and FDE intersect in the points D and D,; then these points correspond to the rising and setting of the sun . Now twilight prevails from sunrise or sunset until the sun is depressed through 18°; hence if we draw arcs ZC and ZC, equal to 108°, and terminating on the circle FDE at C and C,, then the arcs DC and D,C, represent the distance traversed by the sun during the twilight . Also it may be observed that C,EC represents the path of the sun during the night, and DFD1 during the day . The arc CD is readily determined by spherical trigonometry . For, join CP by an arc of a great circle; then in the triangle ZPC we know ZP (the colatitude of O) ; PC (the sun's polar distance) and ZC (=Io8° by construction) . Hence the angle ZPC, the sun's hour angle, may be found; this gives the time before or after noon when the sun passes C . The times of sunrise and sunset being known, then the arcs DC and DIC, (and the duration of dawn and twilight) are determined . So far we have considered the case when the sun does attain a depression of 18°, but it is equally possible for this depression not to be attained . To investigate this, take ZG equal to Io8° . Now if G lies beyond B and E (the maximum depression of the sun), E being also below B, then the sun will rise and set, but never descend so low as to occasion true night, and the entire interval between sunrise and sunset will be twilight . If E be not below B but above it, the sun will never descend below the horizon, and will neither rise nor set, and we are presented with the phenomenon known as the midnight sun .

Since PE=90° — sun's

declination, and PG = latitude of observer + 18°, then it follows that for there to be no night the latitude of the observer together with the declination of the sun must lie between 90° and 72° . The maximum declination of the sun is about 23° 30', and hence in latitude 48° 3o' there will be one day without a true night; in higher latitudes there will be an increasing number of such days; and in lower latitudes none . In England there is no real night from about the 22nd of May till the 22nd of July . The phenomenon known as the after-glow, or second twilight, has been referred to a second reflection of the solar rays in the atmosphere .