INTENSITY OF SUN'S HEAT AND LIGHT. 53 



With this mode of calculation, the first observations of Lambert, before stated, 

 determine the height to be 17 miles; and the second observations, 25 miles. And 

 a still later observation would have given a still greater height, owing, perhaps, to 

 the mingling of direct and reflected rays. The subject awaits further improve- 

 ment ; though some extensions have been made by M. Bravais, in the Annuaire 

 Meteorologique de la France for 1850. 



If we regard only the appearance of the Twilight bow, the limits of the sun's 

 depression assigned by M. Bravais are doubtless nearly correct, namely 16° for 

 astronomical, and 6° for civil twilight. But, regarding only the actual intensity of 

 light falling upon the eye, it appears that the effects of the bow are further 

 increased by indefinite reflection among the particles of air, and this may increase 

 the average limits to 9° for civil, and 18° for astronomical twilight. Without 

 determining which view ought to be adopted, a mean has here been taken, and the 

 following tables have been calculated on the assumption that the sun is 7i° below the 

 horizon at the end of civil twilight ; and 17°, at the end of astronomic twilight. 



The increase of the clay by Refraction and by the twilights, may all be compre- 

 hended in one general formula. Let m denote the sun's depression below the 

 horizon at the end of either period; then the distance from the Pole to the zenith, 

 90° — L, the distance from the Pole to the sun, 90° — D, the distance from the 

 zenith to the sun 90° + m, or three sides of a spherical triangle are given to find 

 the hour angle H -\- T, as in the following equation : — 



,tt , N — sin L sin D — sin m TT sin m , _ _ N 



cos (H + t) = — = + cos II— . (55.) 



cos L cos JJ cos L cos D 



Here <r denotes the increase by refraction or by Twilight, according as m is taken 

 at 34', at 7i°, or 17°. 



When twilight lasts through the whole night, it is evident that at the commence- 

 ment and at the end of such period, 7= 12 h — H. Substituting this value in (55), 



— 1 = —smLsinD — smm,^ Qr .^ B)=sin m; that is,D=90° — L—m. (56.) 

 cos L cos D 



The corresponding yearly limit for constant sunlight has already been found to 

 be indicated by D = 90° — L. The lowest latitude where this is possible is evi- 

 dently L = 90° — a, or at the Polar Circle. In like manner, the lowest latitude 

 where twilight through the whole night occurs, is L = 90° — 1> — m = 49° 32' north 

 or south of the equator. 



During the long night in the Polar regions, twilight will be, for a time, im- 

 possible; that is, so long as the sun continues more than 17° below the horizon. 

 The limits of this period will be defined by making H + r equal to 0, in (55) ; 

 whence L — D = 90° + m, or D = — 90° — m + L. (57.) 



The corresponding yearly limit of sunlight is indicated by D — — 90° + L. But 

 the application of these limits is reserved till after an expression for the annual 

 duration of twilight has been found by the method of summation described in Sec- 

 tion V. For this purpose, equation (55) may be put under the form of 



r = _ H+ cos-' (cos II— S " im - ). (58.) 



\ cos L cos D) 



