362 METEOROLOGY. 



may estimate tliat civil twilight ends, when the sun has declined 6° 

 below the horizon ; and that a decline of 16° is necessary to terminate 

 the astronomic twilight. 



''I depart here i'rom the general opinion, which fixes at 18° the 

 solar depression at the end of twilight, and at 9° that which charac- 

 terizes the end of civil twilight. The numbers which I have adopted 

 are derived from numerous observations." •' The shortest civil twi- 

 light takes place on the 29th of Sej)tember, and on the 15th of March; 

 the longest on the 21st of June. The shortest astronomic t\rilight 

 occurs on the Yth of October, and on the 6th of March ; the longest 

 on the 21st of June, in this latitude. Above the 50th degree of lati- 

 tude twilight lasts through the whole night at the summer solstice." 



The analytic solution of the problem to find the time of the shortest 

 hoilight was first given by John Bernoulli ; the formula may be found 

 in various astronomical works. The method of Lambert for deter- 

 mining the height of the atmosphere from twilight being less com- 

 monly known, a method of solution is given in the Smithsonian 

 Memoir. Lambert found that when the true depression of the sun 

 below the horizon was 8° 03', the height of the twilight arch was 8° 

 30'; and when the depression was 10° 42', the altitude of the bow waa 

 6° 20'. 



With the given mode of calculation, the first observations of Lam- 

 bert determine the height of the atmosphere 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 improvement ; 

 though some extensions have been made by M. Bravais, in the An- 

 nuaire 3Ieteorologique 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 indefi- 

 nite 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 7^° below the horizon at the end of civil 

 twilight, and 17° at (he end of astronomic tioilight. 



By subtracting either value from the latitude of the polar circle we 

 obtain the lowest latitude at which twilight lasts through the whole 

 night at midsummer. This latitude is about 50° for astronomical, 

 and 60° for civil twilight. In determining these and otlier phases, 

 the increase of the day by refraction and by the twilights may all be 

 comprehended in one general formula.* 



• Let m denote thu sim's depression below the horizon at the end of either period ; 

 then the distance from tlie 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° -{- wt, or three sides of a 

 *ph8rical triangle are given to find the hour angle 11 -\- r, as in the following equatiou : 

 ,,, , , — sin L sin D — sin m , ,., sin m 



COS. {II -\- r) = = zz: 4- COS. H ; =• 



* ' cos L COS. D ' COS. L cos. D 



ITero r denotes the increase bj refractioa or bj Twilight, according ;i6 m is taken at 34'. 

 at 7^°, or 170. 



