90 Mr DAVIES on the Nature of the Hour-Lines 



The general equations of the two circles are, 



tanD = tanB < sin. 9 ~ M ' L 

 n, 



tan D s= tan B,, . sin 9 ~"" L 



In which B, B 7/ are the angles at which these circles intersect the equator, 

 and M, are the values of n in the general equation of the hectemoria, 

 adapted to the hour-lines in question. 



Now, if D' be the declination of the parallel, then 



cos"- 1 tan D' cot I 



T it _ cos" 1 tan D' cot I 

 j_, _ 



and hence tan B, = tan D' cosec { 90 - cos"' tan D' cot I 1 



I / j 



tan B. = tan D' cosec I 90 - co^ tan D' cot 1 1 

 I * J 



Which substituted in the equations of the circles, the values of D then 

 equated, and L found, will give the following result : 



.90 90 



c, sin -- r sin - 

 T n. n 



tanL = 90 -- m 



c, cos -- c l cos 



n < 



where c, c lt are put instead of the above named cosecants. 



In the case of the tangents, however, we may more simply take for the 

 values of B /5 B /7 those given in IV. We shall thus get 



.90 .90 



n, sin - -- n sin 



n, cos -- n.. cos 



These values of tan L are functions of n, and , which vary when 

 they vary, and are permanent when they become permanent. Hence, clear- 

 ly, the great circles intersect in points whose co-ordinates L, D are vari- 



