Mr. T. S. Davies on the Problem of Shortest Twilight. 277 



Thirl wall Castle a bed of encrinal limestone reposes on the 

 basalt, and, as is usually the case when in contact with that 

 rock, has become crystalline in texture, and phosphorescent 

 if laid on red-hot iron: its colour is also changed from dark 

 blue to yellowish white. 

 Newcastle-upon-Tyne, Dec. 18, 1832. 



XLVIII. On Bernoulli's Solution of the Problem of Shortest 

 Twilight. ByT. S. Davies, Esq. F.R.S. L. $ E., F.R.A.S.&c. 



[Concluded from p. 185.] 



T>EFORE we proceed, however, to these supplementary 

 •** theorems, we may make one or two other remarks upon 

 the result just obtained. 



By the theory of equations, if p' and p f/ denote the values of 

 p in the last equation, p referring to the minimum, and p" to 

 the maximum cases, we have 



cos p 1 cos p n = cos 2 A (10) 



which is independent of the almacantars between which the 

 star moves. Also taking the difference, 



_2I fl ^-.r.n 8 U 

 COS p — COS p" — cos A 



ft - ™c > C ° S ^ Pl CSD iP ll- COS 2 Pl+ fr l 



cosippop u cosfp, + p y/ 



= cos A Hi + cos P£?.frr- ] - cos Pi± hi} 



^COSftCVD^+COSft + fty) 



= cos A cos Pf SD Pii- cos Pi + Pn 

 cos p,cop u + cos p,+p u 



= cos X tan p / tan p u (11) 



which, when p t and p u (the zenith distances of the almacantars) 

 are taken the complements of each other, the difference of the 

 cosines of the polar distances of the sun for the maximum 

 and minimum time is constant. 



Again, from the same equation (9), we have 



/ n 1 + cos Pi c °s Pi, , i 



coso' + cosp" = 2 cos A — B ^' (12) 



r cos p, + cos p u v ' 



and by division, we have 



cos p' 1 ™**ipp*hi (l3) 



cosp" " cos 2 Ift + pu "'" l 



The question actually resolved is, — what declination of the 



