I 



Solution of Astronomical Problems. 71 



(7) In what direction and at what time will the first signs 

 ot' twilight be observed in north latitude X wdien the 

 sun's declination is 10° S. ? 



On the tracing-cloth trace through the point R coinciding 

 with A, and that one o£ the lines / which is distant 108° from 

 A (or 11). Turn the diagram in the clockwise direction till 

 the angle ROA = 90°— A, and mark on the cloth the point X 

 in wdiich that one of the lines / wdiich is .100° from A intersects 

 the line drawn on the cloth. The angle a between the circles 

 AXB and ACB gives the time of dawn. To find the direction 

 turn the cloth till R coincides with A again and read otf the 

 angle /3 between the circles AXB and ACB, then dawn is 

 seen at an angle f3 from the north point of the horizon. 



For instance, putting X = 50° X., it is found bv the use of 

 AVulfPsnet that «=71°and 3 = 84° 15'; the calculated values 

 are 73° 51' and 84° 4'. 



(8) Trace the changes in the duration of twilight in any 

 given north latitude \ throughout the year. 



AVe shall neglect the change of longitude of the sun during 

 twilight; and shall at first reckon the length of twilight from 

 the time that the zenith distance of the sun is 108° to the 

 time that its true zenith distance (unaffected by refraction) 

 is 90°. 



On the tracing-cloth trace through the point R coinciding 

 with A, P coinciding with (;, the line COD and that one of the 

 lines Z which is distant 108° from A (or R). Turn the diagram 

 in the clockwise direction till the angle ROA = 90°— X. Let 

 one of the lines I distant 7 degrees fronj A cut the two lines 

 traced on the cloth in X and Y ; then the angle between 

 the circles AXB and AYB gives the duration of twilight 

 when the sun's north polar distance is y (the times of daw^n 

 and sunrise are also given) . We can at once find the duration 

 for any value of 7 : thus for latitude 50° N. we find the 

 f ollowinof values for the anole : — 



7=68 



70 



72 



74 



76 



78 80 



82 



84 



^ = 60| 



44 



39 



36 



341 



m 311 



301 



30 



7 = 86 



88 



90 



92 



94 



96 98 



100 



102 



e='2d 



284 



m 



28 



27f 



27A 27i 



27i 



^n 



7 = 104 



106 



108 



110 



112 



degrees. 







^=281 



m 



29 



291 



30i 



degrees. 







If we multiply the numbers in the second rows by 4, we 

 get the length of twilight (expressed in minutes of time) 

 corresponding to the given values of 7. 



