Sohttion of Astronomical Prohlenis. 



73 



value (yo) of 7 for which the sum (180°-f (/>) of the angle 

 between the circles ADB, AXB and the angle between ADB^ 

 AYB is a minimum (X, Y, as before, are the points where 

 any one of the lines / cuts the two lines traced on the cloth). 

 This is the value of y for which daytime reckoned from th(^ 

 middle of twilioht in the morning to the middle of twilioht in 

 the evening is a minimum. From the construction it follows 

 that 



sin (7, - 90°) = sin X. cot 9°; 



hence 72 is real only if sin\> tan 9°, i. ^. if X> 9° 1' (other- 

 wise the point So lies within the circle ABCD), and moreover 

 72 does not he between the limits 90° + i unless X> 3° 27^ 



If X= 2° 20^ 72 = 104° 53', we have graphically the following 

 values of ^ for various values of 7: — 



7=68° 



0=21° 70' 



70° 

 20° 30' 



72° 

 20= 18' 



74° 



19° 59' 



76° 

 19° 38' 



78° 

 19° 14' 



80= 

 19° 0' 



7 = 82= 

 = 18° 36' 



84° 

 18° 12' 



86° 

 18° 8' 



88° 

 18=4' 



90° 



18° 0' 



92° 

 17° 46' 



94° 

 17° 42 



7=96° 



= 17° 38' 



98° 

 17° 34' 



100° 

 17° 31' 



102= 

 17° 28' 



104° 



17° 28' 



106° 

 17° 27' 



108° 

 17° 26 



7 = 110= 



= 17= 30' 



112° 

 17° 30' 



(120°) 

 (18° 0') 



(130°). 

 (19° 25'). 









The minimum value of ^ is not very clearly marked in 

 the tabulated numbers as the graphical readings cannot be 

 trusted to within 10' or so. 



If we wish to determine the interval between dawn and the 



time at which the centre of the sun appears to be on the 



horizon, we trace on the cloth those lines / which are distant 



90 J° and 108° from A (see p. 71, lines 24 et seq.) instead of the 



lines distant 90° and 108° from A (assuming that the sun 



appears to be on the horizon when it is really 30' below). AVe 



then proceed as before, making a few slight modifications. 



For instance, the angles OPA'i, OPYq must now be made 



. , cos 8|° , , , sin 9i° " 



ecjual to tan-^-T^—^^- and tan~^ — Tubo respectively. 



sm 9: 



cos 8J 



The constructions for the points Si and 89 have been based 

 on the following considerations. Firstly, the lines / for 

 which the difference and the sum of the angle between the 

 circles ADB, AXB and the angle between ADB, AYB are 

 minima must cut the lines on the cloth at equal and supple- 

 mentary angles respectively. This is proved readily by 

 geometrical inversion Mith respect to either of the points 

 A or B. Secondly, if the line joining the centres of any two 



