of the Problem of Shortest Twilight, 279 



rjryr, COS — COS A COS 0,, , , . 



cos PZR = r ~ : -' = cot A tan -k (pu—p,) 



sin A sin p u r " r " 



= -cotAtanf(ft_ft y )...(15) 



And by a comparison of (14) (15) we find at once that PZN 

 + PZR = gr, or the azimuths PZN, PZR are supplementary. 



2. Take £/CVft,. Then 



twxi cose — cos A cos p J ■- T/ . ,.^ v 



cos PZN = - r . - . n = cot A tan \ (ft + ft.) ... ( 16) 



sin A sin p, vr ' ru/ v y 



Tir/r* COS — COS A COS ft, _' T/ . ,^ N 



cos PZR = r -.— — -. , r // =cotAtan|(p, + p,,) ...(1?) 



sin A sin ft, 1 vr/ r " v y 



Hence in this case the azimuths are equal ; or the points in 

 which the sun crosses the two almacantars are in the same 

 vertical great circle, — a remarkable property. 



To the case of the shortest twilight problem, the p, + p,i ap- 

 plies, and gives the same value as found by Delambre and 

 others: but as it depends upon the respective values of ft and 

 p n whether the + or cv> designates the minimum time between 

 the almacantars, we cannot say that the supplemental or the 

 equal relation of the azimuths belongs either to the maximum 

 or to the minimum only. 



In the supplementary case we may readily find the differ- 

 ence of the azimuths as follows : — 



Let £, and £ /; denote the azimuths : then by the common 

 properties of angles, we have 



cos fj + cosf,, = 2 cos J ({?, + {?„) cos \ (?,-£„) (a) 



-cos^ + cos Sit = 2 sin i (£, + ?„) sin £ (?,-£„) (b) 



But equation (a) is = 0, and in equation (b) we have 



2 sin \ (£, + £,/) = 1) and hence also 



-cos^ + cos^ == 2sin4({? i -£ fl ) (18) 



But by (14) (15) we have 



— cos £, + cos £» = — 2 cot A tan \ p—pjp and hence 



sin \ (?,-?„) = - cot A tan J P - P// (19) 



In the other case, or that in which azimuths are equal, we 



have also 



sin \ (£,-£„) = cot A tan \ Pi + Pn (20) 



Delambre, after Cagnoli and some other authors, shows 

 that the angles at the sun are equal in the case of a minimum 

 twilight. They are also equal or supplementary in all cases. 



