52 



INTENSITY OF SUN'S HEAT AND LIGHT. 



The numbers which I have adopted are derived from numerous observations." 



"The shortest civil twilight takes place on the 29th of September, and on the 15th 



of March ; the longest on the 21st of June. The shortest astronomic twilight 



occurs on the 7th of October, and on the 6th of March; the longest on the 21st of 



June, in this latitude. Above the 50th degree of latitude, twilight lasts through 



the whole night at the summer solstice." 



The analytic solution of the problem to find the time of shortest twilight was first 



given by John Bernoulli ; the result, found in various works, is expressed by the 



two equations, 



t sin i in 

 sin — = — 

 2 



sin Dec. — — tan i m sin lat. 



cos lat 



Here t denotes the duration of shortest twilight, and m is the sun's depression 16° 

 or 18° below the horizon. 



To pursue the discussion of the physical details of twilight, would occupy too 



much space, and we shall here only 

 glance at the method of Lambert for de- 

 termining the height of the atmosphere 

 from twilight. The demonstration is 

 based upon the examples solved in Geh- 

 ler's Physikalisches Worterbuch. 



Lambert found that when the true 

 depression of the sun below the horizon 

 was 8° 3', (5), the height of the twilight 



arch was 8° 30, (a) ; and when the true 

 ii v J 



depression of the sun was 10° 42', the 



altitude of the bow was 6° 20'. 



In the figure let C denote the centre 



and A B the surface of the earth ; D E, the outer limit of the atmosphere ; B, the 



place of the observer ; and E, the position of the bow. 



Let r denote the refraction due to the altitude (a), then the angle EBB. = a — 

 r = a'. When the sun's centre is apparently in the horizon of A, it is really about 

 34' below it. Denoting this horizontal refraction by f, and deducting it from b, 

 leaves the angle H B8=b — f = b' ; where B 8 is parallel to the tangent A D. 

 But the ray of light after passing A is further refracted by / or 34' to E. 



The angle HB S or &', according to a proposition of geometry, is measured by 

 the arc A B; whence the angle A C B = b' ; then drawing the chord A B, and 

 denoting the earth's radius by R, the isosceles triangle AC B gives AB=2R sin i b'. 



In triangle ABE, the angle A is evidently i b — f ; and deducting the other 

 three angles of the quadrilateral from four right angles, leaves the angle E = 180° 

 — a' — V -\- r 1 ; then, 



sin E ox sin (a' + b' — /) : sm(l b' — f) : : AB : BE = —, 7 , v -. (54.) 



v sm(a'-\-b' — r) 



In the triangle CBE, two sides and the included angle 90° + a' are now given 



to find the third side CE; from which deducting R, leaves the required height of 



the atmosphere. 



