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IX. Tables for finding the Length of Time of Insolation for any 

 Latitude, and for any Day of the Year. 



The formulte for computing the length of time of daily solar illumination are 

 obtained as follows : — 



Let P" be the north pole (celestial), 8 the true place of the 

 sun's centre when that centre is brought by refraction to the hori- 

 zon of the place, and let Z be the zenith of the place. 

 Let 5 = the north declination of the sun 



(negative when south) =90° — PS 



I = the north latitude of the place 



(negative when south) =90° — PZ 



rc= the horizontal refraction = ZS — 90° 



h ;= the hour angle Z P S. 



We have cos ZS = cos PZ cos PS + sin PZ sin PS cos ZPS. 



Or — sin r = sin / sin 5 + cos / cos S cos h [1] 



Also cos (I — 6) :=: cos I cos 6 + «"i ^ sin 8 [2] 



cos (/ -f 6) = cos I cos 5 — sin / sin 6 [3] 



Subtract [1] from [2], and add [1] to [3]. 



cos (/ — 6) + sin r=: cos I cos 8(1 — cos h)z=2 cos I cos 8 sin'' ^h [4] 



cos {I -\- 8) — sin 7- = cos I cos S (1 + cos h) = 2 cos I cos 8 cos* ^h [5] 



Observing that sin a: + sin y= 2 sin ^ (.^ + y) cos ^ {x — i/) the two last equa- 

 tions give us 



sin^ 1 h ^ sin(45°-i/+i8 + iOcos(45°-l^+ig-ir) 

 ^ cos / cos 8 



[7] 



^^,2 1 f, _ sin(4l>°-i/-i8-ir)cos(45°-i/-l8 + ir) 



cos I cos 5 

 which are the formulae used in computing the tables. 



The refraction has been assumed to be 34'. 



The declinations used are from the Nautical Almanac for 1862, for Greenwich 

 mean noon; except in finding the limiting date when the sun's centre does not go 

 below the horizon throughout the whole day, in which case the midnight declination 

 has been used. 



A supplementary table is given by the aid of which the main table may be used 

 for southern as well as northern latitudes. 



The use of the main table may be illustrated by the following example: — 



Find the time of insolation for May 13th, latitude 43°-30 N. 



May 11th, lat. 42° = 14\37 May 11th, lat. 44°= 1 4^54 



I diff. to May 16th, = + .07 f diff. to May 16th = + .07 



May 13th, lat. 42° = 14^44 May 13th, lat. 44°= 14\61 



f diff. 42° to 44° = -f .13 Diff. lat. 42 to 44° = 0M7 



May 13th, lat. 43°-30= 1^^57 ans. 



The use of the supplementary table is sufficiently plain. For example: To find 

 the time of insolation for January 6th in any south latitude, add the tabular number 

 1'*.97 to the corresponding date of July; with the latter and the latitude of the 

 place (regarding it as north instead of south) as arguments, the required time of 

 insolation may at once be found. 



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