72 Mr. H. Hilton on the Graphical 



As might have been expected, it is not easy by tabulating 

 numbers as in the example given to answer exactly the 

 question ''for what value of 7 is the duration of twilight a 

 minimum ? " This problem may, however, be solved graphi- 

 cally with considerable accuracy; but the discussion is rather 

 complicated since 90°^7 must be >/ (the obliquity of the 

 ecliptic), and since only a limited number of the lines I cut 

 both the lines traced on the cloth. 



Make the angle 0PYi = 81°, and let PYi cut RO produced 

 in Vi; draw YiSi perpendicular to OVi cutting AB produced 

 in Si. Then that one of the lines I which passes through the 

 points of contact of tangents from Si to the circle ADBC 

 corresponds to the value (71) of 7 for which ^ is a minimum. 

 It is readily seen from this construction that 



sin (7, -90°)= sin X. tan 9°; 



and since the construction occupies rather a large space, it 

 is preferable to calculate 7] from this formula. For X = 50° 

 we have 7i = 96° 58^ When X = 81°, 7i = 99°, and twilight 

 has a minimum duration of 12 hours ; twilight ceasing 

 exactly at midnight and sunrise taking place exactly at 

 midday. When 7 has any other value the conditions of our 

 problem cannot be satisfied; for if 7>99°, the sun does not 

 rise, and if 7 < 99°, twilight does not cease. If X> 81°, the 

 conditions of our problem are never satisfied. If \< 81°, 

 the value 7 given by the equation sin (7—90°) =sin Xtan 9° 

 always corresponds to a minimum duration of twdlight which 

 is a true minimum satisfying the conditions ; for since 



sinX.tan9°>sin{18°-(90°-\)} if cotX>tan9°, 



twilight ceases and sunrise really takes place when 7 has the 

 value given by the equation 



sin (7 - 90°) = sin X. tan 9°. 



The geometrical construction gives only a minimum and 

 no maxima for duration of twilight ; there must be maxima, 

 however, when 7 has its greatest and smallest possible values 

 {i. e. 90° + 2), if these values satisfy the conditions of the 

 problem. This is the case for the greatest value of 7 in all 

 places in north latitude for which X>90°— /, and for the 

 smallest value of 7 in places for which X>72° — z. 



On the diagram make the angle OPY2 = 9°, and let PY2 

 cut RO produced in Y2, and draw Y2S2 perpendicular to OY2 

 cutting AB (produced if necessary) in S2. Then that one 

 of the lines / which passes through the points of contact of 

 tangents from S2 to the circle ilDBC, corresponds to the 



